Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Percentage Accurate: 57.7% → 97.7%

Time: 15.5s

Alternatives: 22

Speedup: 3.9×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

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)
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
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Python

def code(x, y, z, t, a, b):
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

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)
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
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Python

def code(x, y, z, t, a, b):
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 56000000000:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(15.234687407 \cdot z\right) \cdot z + \mathsf{fma}\left(z \cdot z, z, 31.4690115749 \cdot z\right)\right) + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y z) (- (/ t z) -11.1667541262) (fma 3.13060547623 y x))
          (fma
           (* (/ y (* z z)) -36.52704169880642)
           15.234687407
           (* (/ y z) (- (/ 98.5170599679272 z) -47.69379582500642))))))
   (if (<= z -1e+34)
     t_1
     (if (<= z 56000000000.0)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (*
           (+
            (+ (* (* 15.234687407 z) z) (fma (* z z) z (* 31.4690115749 z)))
            11.9400905721)
           z)
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / z), ((t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(((y / (z * z)) * -36.52704169880642), 15.234687407, ((y / z) * ((98.5170599679272 / z) - -47.69379582500642)));
	double tmp;
	if (z <= -1e+34) {
		tmp = t_1;
	} else if (z <= 56000000000.0) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / ((((((15.234687407 * z) * z) + fma((z * z), z, (31.4690115749 * z))) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / z), Float64(Float64(t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(Float64(Float64(y / Float64(z * z)) * -36.52704169880642), 15.234687407, Float64(Float64(y / z) * Float64(Float64(98.5170599679272 / z) - -47.69379582500642))))
	tmp = 0.0
	if (z <= -1e+34)
		tmp = t_1;
	elseif (z <= 56000000000.0)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(15.234687407 * z) * z) + fma(Float64(z * z), z, Float64(31.4690115749 * z))) + 11.9400905721) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t / z), $MachinePrecision] - -11.1667541262), $MachinePrecision] + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * -36.52704169880642), $MachinePrecision] * 15.234687407 + N[(N[(y / z), $MachinePrecision] * N[(N[(98.5170599679272 / z), $MachinePrecision] - -47.69379582500642), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+34], t$95$1, If[LessEqual[z, 56000000000.0], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(15.234687407 * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] * z + N[(31.4690115749 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999946e33 or 5.6e10 < z

   1. Initial program 10.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)} \]

   if -9.99999999999999946e33 < z < 5.6e10

   1. Initial program 99.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z} + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right)} \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z + \frac{15234687407}{1000000000}\right) \cdot z} + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z + \frac{15234687407}{1000000000}\right)} \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{z \cdot \left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right)} + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      6. distribute-lft-in N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(z \cdot \left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z\right) + z \cdot \frac{314690115749}{10000000000}\right)} + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(z \cdot \left(\color{blue}{\left(\frac{15234687407}{1000000000} + z\right)} \cdot z\right) + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      8. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(z \cdot \color{blue}{\left(z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)} + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      9. distribute-rgt-in N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(z \cdot \color{blue}{\left(\frac{15234687407}{1000000000} \cdot z + z \cdot z\right)} + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      10. unpow2 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(z \cdot \left(\frac{15234687407}{1000000000} \cdot z + \color{blue}{{z}^{2}}\right) + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      11. distribute-rgt-in N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + {z}^{2} \cdot z\right)} + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      12. unpow2 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \color{blue}{\left(z \cdot z\right)} \cdot z\right) + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      13. unpow3 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \color{blue}{{z}^{3}}\right) + z \cdot \frac{314690115749}{10000000000}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      14. *-commutative N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + {z}^{3}\right) + \color{blue}{\frac{314690115749}{10000000000} \cdot z}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      15. associate-+l+ N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \left({z}^{3} + \frac{314690115749}{10000000000} \cdot z\right)\right)} + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      16. lower-+.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \left({z}^{3} + \frac{314690115749}{10000000000} \cdot z\right)\right)} + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      17. lower-*.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z} + \left({z}^{3} + \frac{314690115749}{10000000000} \cdot z\right)\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      18. lower-*.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(\frac{15234687407}{1000000000} \cdot z\right)} \cdot z + \left({z}^{3} + \frac{314690115749}{10000000000} \cdot z\right)\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      19. unpow3 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \left(\color{blue}{\left(z \cdot z\right) \cdot z} + \frac{314690115749}{10000000000} \cdot z\right)\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      20. unpow2 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \left(\color{blue}{{z}^{2}} \cdot z + \frac{314690115749}{10000000000} \cdot z\right)\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      21. lower-fma.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \color{blue}{\mathsf{fma}\left({z}^{2}, z, \frac{314690115749}{10000000000} \cdot z\right)}\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      22. unpow2 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \mathsf{fma}\left(\color{blue}{z \cdot z}, z, \frac{314690115749}{10000000000} \cdot z\right)\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      23. lower-*.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(\frac{15234687407}{1000000000} \cdot z\right) \cdot z + \mathsf{fma}\left(\color{blue}{z \cdot z}, z, \frac{314690115749}{10000000000} \cdot z\right)\right) + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

   3. Applied rewrites 99.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(15.234687407 \cdot z\right) \cdot z + \mathsf{fma}\left(z \cdot z, z, 31.4690115749 \cdot z\right)\right)} + 11.9400905721\right) \cdot z + 0.607771387771} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 56000000000:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\mathsf{fma}\left(z, z, 15.234687407 \cdot z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y z) (- (/ t z) -11.1667541262) (fma 3.13060547623 y x))
          (fma
           (* (/ y (* z z)) -36.52704169880642)
           15.234687407
           (* (/ y z) (- (/ 98.5170599679272 z) -47.69379582500642))))))
   (if (<= z -1e+34)
     t_1
     (if (<= z 56000000000.0)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (*
           (+
            (* (+ (fma z z (* 15.234687407 z)) 31.4690115749) z)
            11.9400905721)
           z)
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / z), ((t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(((y / (z * z)) * -36.52704169880642), 15.234687407, ((y / z) * ((98.5170599679272 / z) - -47.69379582500642)));
	double tmp;
	if (z <= -1e+34) {
		tmp = t_1;
	} else if (z <= 56000000000.0) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((fma(z, z, (15.234687407 * z)) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / z), Float64(Float64(t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(Float64(Float64(y / Float64(z * z)) * -36.52704169880642), 15.234687407, Float64(Float64(y / z) * Float64(Float64(98.5170599679272 / z) - -47.69379582500642))))
	tmp = 0.0
	if (z <= -1e+34)
		tmp = t_1;
	elseif (z <= 56000000000.0)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(fma(z, z, Float64(15.234687407 * z)) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t / z), $MachinePrecision] - -11.1667541262), $MachinePrecision] + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * -36.52704169880642), $MachinePrecision] * 15.234687407 + N[(N[(y / z), $MachinePrecision] * N[(N[(98.5170599679272 / z), $MachinePrecision] - -47.69379582500642), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+34], t$95$1, If[LessEqual[z, 56000000000.0], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(z * z + N[(15.234687407 * z), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999946e33 or 5.6e10 < z

   1. Initial program 10.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)} \]

   if -9.99999999999999946e33 < z < 5.6e10

   1. Initial program 99.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z + \frac{15234687407}{1000000000}\right) \cdot z} + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z + \frac{15234687407}{1000000000}\right)} \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{z \cdot \left(z + \frac{15234687407}{1000000000}\right)} + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. distribute-rgt-in N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z \cdot z + \frac{15234687407}{1000000000} \cdot z\right)} + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\mathsf{fma}\left(z, z, \frac{15234687407}{1000000000} \cdot z\right)} + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      6. lower-*.f64 99.1

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\mathsf{fma}\left(z, z, \color{blue}{15.234687407 \cdot z}\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 99.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\mathsf{fma}\left(z, z, 15.234687407 \cdot z\right)} + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 56000000000:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y z) (- (/ t z) -11.1667541262) (fma 3.13060547623 y x))
          (fma
           (* (/ y (* z z)) -36.52704169880642)
           15.234687407
           (* (/ y z) (- (/ 98.5170599679272 z) -47.69379582500642))))))
   (if (<= z -1e+34)
     t_1
     (if (<= z 56000000000.0)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (+
           0.607771387771
           (* (* (fma (- z -15.234687407) z 31.4690115749) z) z))
          (* 11.9400905721 z))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / z), ((t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(((y / (z * z)) * -36.52704169880642), 15.234687407, ((y / z) * ((98.5170599679272 / z) - -47.69379582500642)));
	double tmp;
	if (z <= -1e+34) {
		tmp = t_1;
	} else if (z <= 56000000000.0) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / ((0.607771387771 + ((fma((z - -15.234687407), z, 31.4690115749) * z) * z)) + (11.9400905721 * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / z), Float64(Float64(t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(Float64(Float64(y / Float64(z * z)) * -36.52704169880642), 15.234687407, Float64(Float64(y / z) * Float64(Float64(98.5170599679272 / z) - -47.69379582500642))))
	tmp = 0.0
	if (z <= -1e+34)
		tmp = t_1;
	elseif (z <= 56000000000.0)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(0.607771387771 + Float64(Float64(fma(Float64(z - -15.234687407), z, 31.4690115749) * z) * z)) + Float64(11.9400905721 * z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t / z), $MachinePrecision] - -11.1667541262), $MachinePrecision] + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * -36.52704169880642), $MachinePrecision] * 15.234687407 + N[(N[(y / z), $MachinePrecision] * N[(N[(98.5170599679272 / z), $MachinePrecision] - -47.69379582500642), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+34], t$95$1, If[LessEqual[z, 56000000000.0], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(0.607771387771 + N[(N[(N[(N[(z - -15.234687407), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + N[(11.9400905721 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999946e33 or 5.6e10 < z

   1. Initial program 10.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)} \]

   if -9.99999999999999946e33 < z < 5.6e10

   1. Initial program 99.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 99.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 56000000000:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z, z, 11.9400905721 \cdot z\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y z) (- (/ t z) -11.1667541262) (fma 3.13060547623 y x))
          (fma
           (* (/ y (* z z)) -36.52704169880642)
           15.234687407
           (* (/ y z) (- (/ 98.5170599679272 z) -47.69379582500642))))))
   (if (<= z -1e+34)
     t_1
     (if (<= z 56000000000.0)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (fma
           (* (fma (- z -15.234687407) z 31.4690115749) z)
           z
           (* 11.9400905721 z))
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / z), ((t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(((y / (z * z)) * -36.52704169880642), 15.234687407, ((y / z) * ((98.5170599679272 / z) - -47.69379582500642)));
	double tmp;
	if (z <= -1e+34) {
		tmp = t_1;
	} else if (z <= 56000000000.0) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (fma((fma((z - -15.234687407), z, 31.4690115749) * z), z, (11.9400905721 * z)) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / z), Float64(Float64(t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(Float64(Float64(y / Float64(z * z)) * -36.52704169880642), 15.234687407, Float64(Float64(y / z) * Float64(Float64(98.5170599679272 / z) - -47.69379582500642))))
	tmp = 0.0
	if (z <= -1e+34)
		tmp = t_1;
	elseif (z <= 56000000000.0)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(fma(Float64(fma(Float64(z - -15.234687407), z, 31.4690115749) * z), z, Float64(11.9400905721 * z)) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t / z), $MachinePrecision] - -11.1667541262), $MachinePrecision] + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * -36.52704169880642), $MachinePrecision] * 15.234687407 + N[(N[(y / z), $MachinePrecision] * N[(N[(98.5170599679272 / z), $MachinePrecision] - -47.69379582500642), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+34], t$95$1, If[LessEqual[z, 56000000000.0], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(z - -15.234687407), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] * z + N[(11.9400905721 * z), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999946e33 or 5.6e10 < z

   1. Initial program 10.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)} \]

   if -9.99999999999999946e33 < z < 5.6e10

   1. Initial program 99.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z} + \frac{607771387771}{1000000000000}} \]

      2. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right)} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z} + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\color{blue}{\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right)} \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      5. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z + \frac{15234687407}{1000000000}\right) \cdot z} + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      6. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\color{blue}{\left(z + \frac{15234687407}{1000000000}\right)} \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{z \cdot \left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right)} + \frac{607771387771}{1000000000000}} \]

      8. distribute-rgt-in N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z\right) \cdot z + \frac{119400905721}{10000000000} \cdot z\right)} + \frac{607771387771}{1000000000000}} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right)} + \frac{607771387771}{1000000000000}} \]

      10. lower-*.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\color{blue}{\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z}, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      11. lower-fma.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z + \frac{15234687407}{1000000000}, z, \frac{314690115749}{10000000000}\right)} \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      12. metadata-eval N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z + \color{blue}{\frac{15234687407}{1000000000} \cdot 1}, z, \frac{314690115749}{10000000000}\right) \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z - \left(\mathsf{neg}\left(\frac{15234687407}{1000000000}\right)\right) \cdot 1}, z, \frac{314690115749}{10000000000}\right) \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      14. metadata-eval N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z - \color{blue}{\frac{-15234687407}{1000000000}} \cdot 1, z, \frac{314690115749}{10000000000}\right) \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      15. metadata-eval N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z - \color{blue}{\frac{-15234687407}{1000000000}}, z, \frac{314690115749}{10000000000}\right) \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      16. lower--.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z - \frac{-15234687407}{1000000000}}, z, \frac{314690115749}{10000000000}\right) \cdot z, z, \frac{119400905721}{10000000000} \cdot z\right) + \frac{607771387771}{1000000000000}} \]

      17. lower-*.f64 99.1

          \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z, z, \color{blue}{11.9400905721 \cdot z}\right) + 0.607771387771} \]

   3. Applied rewrites 99.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z, z, 11.9400905721 \cdot z\right)} + 0.607771387771} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 56000000000:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y z) (- (/ t z) -11.1667541262) (fma 3.13060547623 y x))
          (fma
           (* (/ y (* z z)) -36.52704169880642)
           15.234687407
           (* (/ y z) (- (/ 98.5170599679272 z) -47.69379582500642))))))
   (if (<= z -1e+34)
     t_1
     (if (<= z 56000000000.0)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (*
           (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
           z)
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / z), ((t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(((y / (z * z)) * -36.52704169880642), 15.234687407, ((y / z) * ((98.5170599679272 / z) - -47.69379582500642)));
	double tmp;
	if (z <= -1e+34) {
		tmp = t_1;
	} else if (z <= 56000000000.0) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / z), Float64(Float64(t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(Float64(Float64(y / Float64(z * z)) * -36.52704169880642), 15.234687407, Float64(Float64(y / z) * Float64(Float64(98.5170599679272 / z) - -47.69379582500642))))
	tmp = 0.0
	if (z <= -1e+34)
		tmp = t_1;
	elseif (z <= 56000000000.0)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t / z), $MachinePrecision] - -11.1667541262), $MachinePrecision] + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * -36.52704169880642), $MachinePrecision] * 15.234687407 + N[(N[(y / z), $MachinePrecision] * N[(N[(98.5170599679272 / z), $MachinePrecision] - -47.69379582500642), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+34], t$95$1, If[LessEqual[z, 56000000000.0], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999946e33 or 5.6e10 < z

   1. Initial program 10.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)} \]

   if -9.99999999999999946e33 < z < 5.6e10

   1. Initial program 99.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)\\ \mathbf{if}\;z \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 11:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y z) (- (/ t z) -11.1667541262) (fma 3.13060547623 y x))
          (fma
           (* (/ y (* z z)) -36.52704169880642)
           15.234687407
           (* (/ y z) (- (/ 98.5170599679272 z) -47.69379582500642))))))
   (if (<= z -0.05)
     t_1
     (if (<= z 11.0)
       (+
        x
        (/ (* y (fma (fma t z a) z b)) (+ (* 11.9400905721 z) 0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / z), ((t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(((y / (z * z)) * -36.52704169880642), 15.234687407, ((y / z) * ((98.5170599679272 / z) - -47.69379582500642)));
	double tmp;
	if (z <= -0.05) {
		tmp = t_1;
	} else if (z <= 11.0) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / z), Float64(Float64(t / z) - -11.1667541262), fma(3.13060547623, y, x)) - fma(Float64(Float64(y / Float64(z * z)) * -36.52704169880642), 15.234687407, Float64(Float64(y / z) * Float64(Float64(98.5170599679272 / z) - -47.69379582500642))))
	tmp = 0.0
	if (z <= -0.05)
		tmp = t_1;
	elseif (z <= 11.0)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * N[(N[(t / z), $MachinePrecision] - -11.1667541262), $MachinePrecision] + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * -36.52704169880642), $MachinePrecision] * 15.234687407 + N[(N[(y / z), $MachinePrecision] * N[(N[(98.5170599679272 / z), $MachinePrecision] - -47.69379582500642), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.05], t$95$1, If[LessEqual[z, 11.0], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.050000000000000003 or 11 < z

   1. Initial program 16.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - -11.1667541262, \mathsf{fma}\left(3.13060547623, y, x\right)\right) - \mathsf{fma}\left(\frac{y}{z \cdot z} \cdot -36.52704169880642, 15.234687407, \frac{y}{z} \cdot \left(\frac{98.5170599679272}{z} - -47.69379582500642\right)\right)} \]

   if -0.050000000000000003 < z < 11

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 79.8%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{119400905721}{10000000000}} \cdot z + \frac{607771387771}{1000000000000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.6%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{11.9400905721} \cdot z + 0.607771387771} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(a + t \cdot z\right) + \color{blue}{b}\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(a + t \cdot z\right) \cdot z + b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(a + t \cdot z, \color{blue}{z}, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z + a, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      10. lower-fma.f64 98.5

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   10. Applied rewrites 98.5%

       \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{11.9400905721 \cdot z + 0.607771387771} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}\\ t_2 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+227}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b)
          (/
           y
           (fma
            (fma (fma (- z -15.234687407) z 31.4690115749) z 11.9400905721)
            z
            0.607771387771))))
        (t_2
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_2 -1e+215)
     t_1
     (if (<= t_2 2e+227)
       (+
        x
        (/ (* y (fma (fma t z a) z b)) (+ (* 11.9400905721 z) 0.607771387771)))
       (if (<= t_2 INFINITY) t_1 (- x (* -3.13060547623 y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) * (y / fma(fma(fma((z - -15.234687407), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771));
	double t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_2 <= -1e+215) {
		tmp = t_1;
	} else if (t_2 <= 2e+227) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) * Float64(y / fma(fma(fma(Float64(z - -15.234687407), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)))
	t_2 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -1e+215)
		tmp = t_1;
	elseif (t_2 <= 2e+227)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] * N[(y / N[(N[(N[(N[(z - -15.234687407), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+215], t$95$1, If[LessEqual[t$95$2, 2e+227], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -9.99999999999999907e214 or 2.0000000000000002e227 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 81.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   if -9.99999999999999907e214 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 2.0000000000000002e227

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 80.9%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{119400905721}{10000000000}} \cdot z + \frac{607771387771}{1000000000000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.7%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{11.9400905721} \cdot z + 0.607771387771} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(a + t \cdot z\right) + \color{blue}{b}\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(a + t \cdot z\right) \cdot z + b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(a + t \cdot z, \color{blue}{z}, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z + a, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      10. lower-fma.f64 90.6

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   10. Applied rewrites 90.6%

       \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{11.9400905721 \cdot z + 0.607771387771} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 81.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 81.3%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 14.2

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 14.2

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 14.2%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{0.607771387771}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.55e+47)
   (fma 3.13060547623 y x)
   (if (<= z 1.6e-7)
     (+
      x
      (/
       (*
        y
        (+
         (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
         b))
       0.607771387771))
     (if (<= z 4.6e+55)
       (fma
        y
        (*
         z
         (/
          (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a)
          (fma
           (fma (fma (- z -15.234687407) z 31.4690115749) z 11.9400905721)
           z
           0.607771387771)))
        x)
       (- x (* -3.13060547623 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.55e+47) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 1.6e-7) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / 0.607771387771);
	} else if (z <= 4.6e+55) {
		tmp = fma(y, (z * (fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a) / fma(fma(fma((z - -15.234687407), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771))), x);
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.55e+47)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 1.6e-7)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / 0.607771387771));
	elseif (z <= 4.6e+55)
		tmp = fma(y, Float64(z * Float64(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a) / fma(fma(fma(Float64(z - -15.234687407), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771))), x);
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.55e+47], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 1.6e-7], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e+55], N[(y * N[(z * N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] / N[(N[(N[(N[(z - -15.234687407), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.55e47

   1. Initial program 5.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 94.3

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -1.55e47 < z < 1.6e-7

   1. Initial program 98.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\frac{607771387771}{1000000000000}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 94.6%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{0.607771387771}} \]

   if 1.6e-7 < z < 4.59999999999999975e55

   1. Initial program 77.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)} \]

   if 4.59999999999999975e55 < z

   1. Initial program 3.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 3.6%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 95.4

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 95.4

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 95.4%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left(\left(z \cdot z\right) \cdot z\right) \cdot z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (+
          x
          (/
           (*
            y
            (+
             (*
              (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a)
              z)
             b))
           (+
            (*
             (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
             z)
            0.607771387771)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b)
      (/ y (* (* (* z z) z) z)))
     (if (<= t_1 INFINITY)
       (+
        x
        (/ (* y (fma (fma t z a) z b)) (+ (* 11.9400905721 z) 0.607771387771)))
       (- x (* -3.13060547623 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) * (y / (((z * z) * z) * z));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) * Float64(y / Float64(Float64(Float64(z * z) * z) * z)));
	elseif (t_1 <= Inf)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] * N[(y / N[(N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))) < -inf.0

   1. Initial program 72.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{\color{blue}{4}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{4}} \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{4}} \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{4}} \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{\left(3 + 1\right)}} \]

      6. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{3} \cdot z} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{{z}^{3} \cdot z} \]

      8. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left(\left(z \cdot z\right) \cdot z\right) \cdot z} \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left({z}^{2} \cdot z\right) \cdot z} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left({z}^{2} \cdot z\right) \cdot z} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left(\left(z \cdot z\right) \cdot z\right) \cdot z} \]

      12. lower-*.f64 84.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left(\left(z \cdot z\right) \cdot z\right) \cdot z} \]

   7. Applied rewrites 84.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\left(\left(z \cdot z\right) \cdot z\right) \cdot \color{blue}{z}} \]

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))) < +inf.0

   1. Initial program 96.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 76.0%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{119400905721}{10000000000}} \cdot z + \frac{607771387771}{1000000000000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{11.9400905721} \cdot z + 0.607771387771} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(a + t \cdot z\right) + \color{blue}{b}\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(a + t \cdot z\right) \cdot z + b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(a + t \cdot z, \color{blue}{z}, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z + a, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      10. lower-fma.f64 88.3

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   10. Applied rewrites 88.3%

       \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{11.9400905721 \cdot z + 0.607771387771} \]

   if +inf.0 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 0.0%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 97.0

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 97.0

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 97.0%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 31000:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.55e+47)
   (fma 3.13060547623 y x)
   (if (<= z 31000.0)
     (+
      x
      (/
       (*
        y
        (+
         (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
         b))
       0.607771387771))
     (fma (/ y z) -36.52704169880642 (fma 3.13060547623 y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.55e+47) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 31000.0) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / 0.607771387771);
	} else {
		tmp = fma((y / z), -36.52704169880642, fma(3.13060547623, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.55e+47)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 31000.0)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / 0.607771387771));
	else
		tmp = fma(Float64(y / z), -36.52704169880642, fma(3.13060547623, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.55e+47], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 31000.0], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -36.52704169880642 + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e47

   1. Initial program 5.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 94.3

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -1.55e47 < z < 31000

   1. Initial program 98.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\frac{607771387771}{1000000000000}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 94.2%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{0.607771387771}} \]

   if 31000 < z

   1. Initial program 14.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right)\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto x + \color{blue}{\left(\left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right)} \]

      2. associate--l+ N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{\left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right)}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \color{blue}{\left(\frac{55833770631}{5000000000} - \frac{4769379582500641883561}{100000000000000000000}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \frac{-3652704169880641883561}{100000000000000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \frac{\frac{3652704169880641883561}{100000000000000000000}}{\color{blue}{-1}}\right) \]

      6. metadata-eval N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \frac{\frac{-55833770631}{5000000000} - \frac{-4769379582500641883561}{100000000000000000000}}{-1}\right) \]

      7. times-frac N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y \cdot \left(\frac{-55833770631}{5000000000} - \frac{-4769379582500641883561}{100000000000000000000}\right)}{\color{blue}{z \cdot -1}}\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{\color{blue}{z} \cdot -1}\right) \]

      9. *-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{-1 \cdot \color{blue}{z}}\right) \]

      10. mul-1-neg N/A

          \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{\mathsf{neg}\left(z\right)}\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\mathsf{neg}\left(\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + -1 \cdot \color{blue}{\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      13. +-commutative N/A

          \[\leadsto x + \left(-1 \cdot \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \color{blue}{\frac{313060547623}{100000000000} \cdot y}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(-1 \cdot \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right) + \color{blue}{x} \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, \mathsf{fma}\left(3.13060547623, y, x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 31000:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.05)
   (fma 3.13060547623 y x)
   (if (<= z 31000.0)
     (+
      x
      (/ (* y (fma (fma t z a) z b)) (+ (* 11.9400905721 z) 0.607771387771)))
     (fma (/ y z) -36.52704169880642 (fma 3.13060547623 y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.05) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 31000.0) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else {
		tmp = fma((y / z), -36.52704169880642, fma(3.13060547623, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.05)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 31000.0)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	else
		tmp = fma(Float64(y / z), -36.52704169880642, fma(3.13060547623, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.05], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 31000.0], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -36.52704169880642 + N[(3.13060547623 * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.050000000000000003

   1. Initial program 17.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 86.5

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -0.050000000000000003 < z < 31000

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 79.6%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{119400905721}{10000000000}} \cdot z + \frac{607771387771}{1000000000000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.4%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{11.9400905721} \cdot z + 0.607771387771} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(a + t \cdot z\right) + \color{blue}{b}\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(a + t \cdot z\right) \cdot z + b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(a + t \cdot z, \color{blue}{z}, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z + a, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      10. lower-fma.f64 98.3

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   10. Applied rewrites 98.3%

       \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{11.9400905721 \cdot z + 0.607771387771} \]

   if 31000 < z

   1. Initial program 14.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right)\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto x + \color{blue}{\left(\left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right)} \]

      2. associate--l+ N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{\left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right)}\right) \]

      3. distribute-rgt-out-- N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \color{blue}{\left(\frac{55833770631}{5000000000} - \frac{4769379582500641883561}{100000000000000000000}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \frac{-3652704169880641883561}{100000000000000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \frac{\frac{3652704169880641883561}{100000000000000000000}}{\color{blue}{-1}}\right) \]

      6. metadata-eval N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y}{z} \cdot \frac{\frac{-55833770631}{5000000000} - \frac{-4769379582500641883561}{100000000000000000000}}{-1}\right) \]

      7. times-frac N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{y \cdot \left(\frac{-55833770631}{5000000000} - \frac{-4769379582500641883561}{100000000000000000000}\right)}{\color{blue}{z \cdot -1}}\right) \]

      8. distribute-rgt-out-- N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{\color{blue}{z} \cdot -1}\right) \]

      9. *-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{-1 \cdot \color{blue}{z}}\right) \]

      10. mul-1-neg N/A

          \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{\mathsf{neg}\left(z\right)}\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\mathsf{neg}\left(\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + -1 \cdot \color{blue}{\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      13. +-commutative N/A

          \[\leadsto x + \left(-1 \cdot \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \color{blue}{\frac{313060547623}{100000000000} \cdot y}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(-1 \cdot \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right) + \color{blue}{x} \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, \mathsf{fma}\left(3.13060547623, y, x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-187}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, \mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_1 -1e+91)
     (* (fma (fma t z a) z b) (* 1.6453555072203998 y))
     (if (<= t_1 -1e-187)
       (fma
        (* y (fma 1.6453555072203998 a (* -32.324150453290734 b)))
        z
        (fma (* b y) 1.6453555072203998 x))
       (if (<= t_1 INFINITY)
         (+
          x
          (/ (* y (fma (* t z) z b)) (+ (* 11.9400905721 z) 0.607771387771)))
         (- x (* -3.13060547623 y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_1 <= -1e+91) {
		tmp = fma(fma(t, z, a), z, b) * (1.6453555072203998 * y);
	} else if (t_1 <= -1e-187) {
		tmp = fma((y * fma(1.6453555072203998, a, (-32.324150453290734 * b))), z, fma((b * y), 1.6453555072203998, x));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x + ((y * fma((t * z), z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -1e+91)
		tmp = Float64(fma(fma(t, z, a), z, b) * Float64(1.6453555072203998 * y));
	elseif (t_1 <= -1e-187)
		tmp = fma(Float64(y * fma(1.6453555072203998, a, Float64(-32.324150453290734 * b))), z, fma(Float64(b * y), 1.6453555072203998, x));
	elseif (t_1 <= Inf)
		tmp = Float64(x + Float64(Float64(y * fma(Float64(t * z), z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+91], N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] * N[(1.6453555072203998 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-187], N[(N[(y * N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x + N[(N[(y * N[(N[(t * z), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1.00000000000000008e91

   1. Initial program 87.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot \color{blue}{y}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 72.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   7. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot \color{blue}{y}\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   if -1.00000000000000008e91 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1e-187

   1. Initial program 99.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)\right) + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto \left(z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right) + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) + x \]

      3. associate-+l+ N/A

         \[\leadsto z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right) + \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right) \cdot z + \left(\color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} + x\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right) \cdot z + \left(x + \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right), \color{blue}{z}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, \mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\right)} \]

   if -1e-187 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 94.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 76.7%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{119400905721}{10000000000}} \cdot z + \frac{607771387771}{1000000000000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.1%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{11.9400905721} \cdot z + 0.607771387771} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(a + t \cdot z\right) + \color{blue}{b}\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(a + t \cdot z\right) \cdot z + b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(a + t \cdot z, \color{blue}{z}, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z + a, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      10. lower-fma.f64 87.5

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   10. Applied rewrites 87.5%

       \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{11.9400905721 \cdot z + 0.607771387771} \]

   11. Taylor expanded in z around inf

       \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   12. Step-by-step derivation

       1. lower-*.f64 78.8

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   13. Applied rewrites 78.8%

       \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 0.0%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 97.0

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 97.0

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 97.0%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_1 -4e+177)
     (* (fma (fma t z a) z b) (* 1.6453555072203998 y))
     (if (<= t_1 INFINITY)
       (+ x (/ (* y (fma (* t z) z b)) (+ (* 11.9400905721 z) 0.607771387771)))
       (- x (* -3.13060547623 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_1 <= -4e+177) {
		tmp = fma(fma(t, z, a), z, b) * (1.6453555072203998 * y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x + ((y * fma((t * z), z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -4e+177)
		tmp = Float64(fma(fma(t, z, a), z, b) * Float64(1.6453555072203998 * y));
	elseif (t_1 <= Inf)
		tmp = Float64(x + Float64(Float64(y * fma(Float64(t * z), z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+177], N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] * N[(1.6453555072203998 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x + N[(N[(y * N[(N[(t * z), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -4e177

   1. Initial program 83.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot \color{blue}{y}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 75.5

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   7. Applied rewrites 75.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot \color{blue}{y}\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 75.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   if -4e177 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 96.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 77.3%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{119400905721}{10000000000}} \cdot z + \frac{607771387771}{1000000000000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.3%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{11.9400905721} \cdot z + 0.607771387771} \]

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      2. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(b + z \cdot \left(a + t \cdot z\right)\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      6. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(a + t \cdot z\right) + \color{blue}{b}\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      7. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\left(a + t \cdot z\right) \cdot z + b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(a + t \cdot z, \color{blue}{z}, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      9. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z + a, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]

      10. lower-fma.f64 87.9

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   10. Applied rewrites 87.9%

       \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}}{11.9400905721 \cdot z + 0.607771387771} \]

   11. Taylor expanded in z around inf

       \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{\frac{119400905721}{10000000000} \cdot z + \frac{607771387771}{1000000000000}} \]
   12. Step-by-step derivation

       1. lower-*.f64 78.9

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   13. Applied rewrites 78.9%

       \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{11.9400905721 \cdot z + 0.607771387771} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 0.0%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 97.0

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 97.0

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 97.0%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x + \frac{y \cdot b}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_1 -5e+89)
     (* (fma (fma t z a) z b) (* 1.6453555072203998 y))
     (if (<= t_1 INFINITY)
       (+ x (/ (* y b) (fma 11.9400905721 z 0.607771387771)))
       (- x (* -3.13060547623 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_1 <= -5e+89) {
		tmp = fma(fma(t, z, a), z, b) * (1.6453555072203998 * y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x + ((y * b) / fma(11.9400905721, z, 0.607771387771));
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -5e+89)
		tmp = Float64(fma(fma(t, z, a), z, b) * Float64(1.6453555072203998 * y));
	elseif (t_1 <= Inf)
		tmp = Float64(x + Float64(Float64(y * b) / fma(11.9400905721, z, 0.607771387771)));
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+89], N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] * N[(1.6453555072203998 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x + N[(N[(y * b), $MachinePrecision] / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -4.99999999999999983e89

   1. Initial program 87.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot \color{blue}{y}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 72.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   7. Applied rewrites 72.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot \color{blue}{y}\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   if -4.99999999999999983e89 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 95.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 77.6%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{607771387771}{1000000000000} + \frac{119400905721}{10000000000} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\frac{119400905721}{10000000000} \cdot z + \color{blue}{\frac{607771387771}{1000000000000}}} \]

      2. lower-fma.f64 75.6

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(11.9400905721, \color{blue}{z}, 0.607771387771\right)} \]

   7. Applied rewrites 75.6%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 0.0%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 97.0

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 97.0

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 97.0%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(a, z, b\right) \cdot \left(1.6453555072203998 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x + \frac{y \cdot b}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_1 -4e+177)
     (* (fma a z b) (* 1.6453555072203998 y))
     (if (<= t_1 INFINITY)
       (+ x (/ (* y b) (fma 11.9400905721 z 0.607771387771)))
       (- x (* -3.13060547623 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_1 <= -4e+177) {
		tmp = fma(a, z, b) * (1.6453555072203998 * y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x + ((y * b) / fma(11.9400905721, z, 0.607771387771));
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -4e+177)
		tmp = Float64(fma(a, z, b) * Float64(1.6453555072203998 * y));
	elseif (t_1 <= Inf)
		tmp = Float64(x + Float64(Float64(y * b) / fma(11.9400905721, z, 0.607771387771)));
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+177], N[(N[(a * z + b), $MachinePrecision] * N[(1.6453555072203998 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x + N[(N[(y * b), $MachinePrecision] / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -4e177

   1. Initial program 83.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot \color{blue}{y}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 75.5

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   7. Applied rewrites 75.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot \color{blue}{y}\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(a, z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.2%

       \[\leadsto \mathsf{fma}\left(a, z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   if -4e177 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 96.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

   4. Applied rewrites 77.3%

      \[\leadsto x + \frac{y \cdot \color{blue}{b}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\frac{607771387771}{1000000000000} + \frac{119400905721}{10000000000} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\frac{119400905721}{10000000000} \cdot z + \color{blue}{\frac{607771387771}{1000000000000}}} \]

      2. lower-fma.f64 75.3

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(11.9400905721, \color{blue}{z}, 0.607771387771\right)} \]

   7. Applied rewrites 75.3%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 0.0%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 97.0

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 97.0

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 97.0%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(a, z, b\right) \cdot \left(1.6453555072203998 \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_1 -1e+215)
     (* (fma a z b) (* 1.6453555072203998 y))
     (if (<= t_1 INFINITY)
       (fma (* b y) 1.6453555072203998 x)
       (- x (* -3.13060547623 y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_1 <= -1e+215) {
		tmp = fma(a, z, b) * (1.6453555072203998 * y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -1e+215)
		tmp = Float64(fma(a, z, b) * Float64(1.6453555072203998 * y));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+215], N[(N[(a * z + b), $MachinePrecision] * N[(1.6453555072203998 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -9.99999999999999907e214

   1. Initial program 81.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot \color{blue}{y}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 76.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   7. Applied rewrites 76.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \left(1.6453555072203998 \cdot \color{blue}{y}\right) \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(a, z, b\right) \cdot \left(\frac{1000000000000}{607771387771} \cdot y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.4%

       \[\leadsto \mathsf{fma}\left(a, z, b\right) \cdot \left(1.6453555072203998 \cdot y\right) \]

   if -9.99999999999999907e214 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 96.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot y\right) \cdot \frac{1000000000000}{607771387771} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot y, \color{blue}{\frac{1000000000000}{607771387771}}, x\right) \]

      4. lower-*.f64 74.7

         \[\leadsto \mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right) \]

   4. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 0.0%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 97.0

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 97.0

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 97.0%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 82.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -480:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 28000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -480.0)
   (fma 3.13060547623 y x)
   (if (<= z 28000.0)
     (fma (* b y) 1.6453555072203998 x)
     (- x (* -3.13060547623 y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -480.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 28000.0) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -480.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 28000.0)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -480.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 28000.0], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -480

   1. Initial program 16.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 87.2

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -480 < z < 28000

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot y\right) \cdot \frac{1000000000000}{607771387771} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot y, \color{blue}{\frac{1000000000000}{607771387771}}, x\right) \]

      4. lower-*.f64 79.1

         \[\leadsto \mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right) \]

   4. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]

   if 28000 < z

   1. Initial program 14.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 14.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 88.6

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 88.6

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 88.6%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 72.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1.6453555072203998 \cdot \left(b \cdot y\right)\\ t_2 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -8.5 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+122}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 1.6453555072203998 (* b y)))
        (t_2
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_2 -8.5e+33)
     t_1
     (if (<= t_2 5e+122)
       x
       (if (<= t_2 INFINITY) t_1 (- x (* -3.13060547623 y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.6453555072203998 * (b * y);
	double t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_2 <= -8.5e+33) {
		tmp = t_1;
	} else if (t_2 <= 5e+122) {
		tmp = x;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.6453555072203998 * (b * y);
	double t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_2 <= -8.5e+33) {
		tmp = t_1;
	} else if (t_2 <= 5e+122) {
		tmp = x;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.6453555072203998 * (b * y)
	t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)
	tmp = 0
	if t_2 <= -8.5e+33:
		tmp = t_1
	elif t_2 <= 5e+122:
		tmp = x
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = x - (-3.13060547623 * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.6453555072203998 * Float64(b * y))
	t_2 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -8.5e+33)
		tmp = t_1;
	elseif (t_2 <= 5e+122)
		tmp = x;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.6453555072203998 * (b * y);
	t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	tmp = 0.0;
	if (t_2 <= -8.5e+33)
		tmp = t_1;
	elseif (t_2 <= 5e+122)
		tmp = x;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = x - (-3.13060547623 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.6453555072203998 * N[(b * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -8.5e+33], t$95$1, If[LessEqual[t$95$2, 5e+122], x, If[LessEqual[t$95$2, Infinity], t$95$1, N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -8.4999999999999998e33 or 4.99999999999999989e122 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 88.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{1000000000000}{607771387771} \cdot \color{blue}{\left(b \cdot y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.7%

      \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(b \cdot y\right)} \]

   if -8.4999999999999998e33 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 4.99999999999999989e122

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 66.2%

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

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 88.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 88.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 19.8

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 19.8

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 19.8%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 64.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - -3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e-23)
   (fma 3.13060547623 y x)
   (if (<= z 1.8e-66) x (- x (* -3.13060547623 y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e-23) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 1.8e-66) {
		tmp = x;
	} else {
		tmp = x - (-3.13060547623 * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e-23)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 1.8e-66)
		tmp = x;
	else
		tmp = Float64(x - Float64(-3.13060547623 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e-23], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 1.8e-66], x, N[(x - N[(-3.13060547623 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8500000000000001e-23

   1. Initial program 23.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 81.9

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   4. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -1.8500000000000001e-23 < z < 1.80000000000000006e-66

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 43.3%

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

   if 1.80000000000000006e-66 < z

   1. Initial program 29.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   3. Applied rewrites 29.1%

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(0.607771387771 + \left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right) \cdot z\right) \cdot z\right) + 11.9400905721 \cdot z}} \]

   4. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 78.1

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   6. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. +-commutative N/A

         \[\leadsto x + \color{blue}{\frac{313060547623}{100000000000} \cdot y} \]

      3. *-commutative N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{313060547623}{100000000000}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto x - \left(\mathsf{neg}\left(y \cdot \frac{313060547623}{100000000000}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000} \cdot y\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{313060547623}{100000000000}\right)\right) \cdot \color{blue}{y} \]

      9. metadata-eval N/A

         \[\leadsto x - \frac{-313060547623}{100000000000} \cdot y \]

      10. lower-*.f64 78.1

          \[\leadsto x - -3.13060547623 \cdot \color{blue}{y} \]

   8. Applied rewrites 78.1%

      \[\leadsto x - \color{blue}{-3.13060547623 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 64.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.85e-23)
   (fma 3.13060547623 y x)
   (if (<= z 1.8e-66) x (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.85e-23) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 1.8e-66) {
		tmp = x;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.85e-23)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 1.8e-66)
		tmp = x;
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.85e-23], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 1.8e-66], x, N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8500000000000001e-23 or 1.80000000000000006e-66 < z

   1. Initial program 26.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{313060547623}{100000000000} \cdot y + \color{blue}{x} \]

      2. lower-fma.f64 79.9

         \[\leadsto \mathsf{fma}\left(3.13060547623, \color{blue}{y}, x\right) \]

   4. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]

   if -1.8500000000000001e-23 < z < 1.80000000000000006e-66

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 43.3%

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

Alternative 21: 49.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.25 \cdot 10^{+130}:\\ \;\;\;\;3.13060547623 \cdot y\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;3.13060547623 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.25e+130)
   (* 3.13060547623 y)
   (if (<= y 45.0) x (* 3.13060547623 y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.25e+130) {
		tmp = 3.13060547623 * y;
	} else if (y <= 45.0) {
		tmp = x;
	} else {
		tmp = 3.13060547623 * y;
	}
	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)
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) :: tmp
    if (y <= (-4.25d+130)) then
        tmp = 3.13060547623d0 * y
    else if (y <= 45.0d0) then
        tmp = x
    else
        tmp = 3.13060547623d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.25e+130) {
		tmp = 3.13060547623 * y;
	} else if (y <= 45.0) {
		tmp = x;
	} else {
		tmp = 3.13060547623 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.25e+130:
		tmp = 3.13060547623 * y
	elif y <= 45.0:
		tmp = x
	else:
		tmp = 3.13060547623 * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.25e+130)
		tmp = Float64(3.13060547623 * y);
	elseif (y <= 45.0)
		tmp = x;
	else
		tmp = Float64(3.13060547623 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.25e+130)
		tmp = 3.13060547623 * y;
	elseif (y <= 45.0)
		tmp = x;
	else
		tmp = 3.13060547623 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.25e+130], N[(3.13060547623 * y), $MachinePrecision], If[LessEqual[y, 45.0], x, N[(3.13060547623 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.24999999999999982e130 or 45 < y

   1. Initial program 55.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(b + z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{313060547623}{100000000000} \cdot \color{blue}{y} \]
   6. Step-by-step derivation

   7. Applied rewrites 34.0%

      \[\leadsto 3.13060547623 \cdot \color{blue}{y} \]

   if -4.24999999999999982e130 < y < 45

   1. Initial program 59.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 60.4%

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

Alternative 22: 43.9% accurate, 52.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	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)
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
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

   \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

2. Taylor expanded in x around inf

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

4. Applied rewrites 43.9%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64
  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))