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

Percentage Accurate: 58.6% → 98.5%

Time: 6.7s

Alternatives: 15

Speedup: 8.8×

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: 58.6% 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: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(-15.234687407, 457.9610022158428 + t, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          y
          (+
           (-
            (/
             (+
              (-
               (/
                (+
                 (+
                  (-
                   (/
                    (-
                     (- a)
                     (fma
                      -15.234687407
                      (+ 457.9610022158428 t)
                      1112.0901850848957))
                    z))
                  t)
                 457.9610022158428)
                z))
              36.52704169880642)
             z))
           3.13060547623)
          x)))
   (if (<= z -17000000.0)
     t_1
     (if (<= z 3.9e+31)
       (+
        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, (-((-(((-((-a - fma(-15.234687407, (457.9610022158428 + t), 1112.0901850848957)) / z) + t) + 457.9610022158428) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	double tmp;
	if (z <= -17000000.0) {
		tmp = t_1;
	} else if (z <= 3.9e+31) {
		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 = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-a) - fma(-15.234687407, Float64(457.9610022158428 + t), 1112.0901850848957)) / z)) + t) + 457.9610022158428) / z)) + 36.52704169880642) / z)) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = t_1;
	elseif (z <= 3.9e+31)
		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[(y * N[((-N[(N[((-N[(N[(N[((-N[(N[((-a) - N[(-15.234687407 * N[(457.9610022158428 + t), $MachinePrecision] + 1112.0901850848957), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision] + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -17000000.0], t$95$1, If[LessEqual[z, 3.9e+31], 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 < -1.7e7 or 3.89999999999999999e31 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + \left(t + -1 \cdot \frac{-1 \cdot a - \left(\frac{1112090185084895700201045470302189}{1000000000000000000000000000000} + \frac{-15234687407}{1000000000} \cdot \left(\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t\right)\right)}{z}\right)}{z}}{z}}, x\right) \]

   9. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(-15.234687407, 457.9610022158428 + t, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]

   if -1.7e7 < z < 3.89999999999999999e31

   1. Initial program 58.6%

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

Alternative 2: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(-15.234687407, 457.9610022158428 + t, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          y
          (+
           (-
            (/
             (+
              (-
               (/
                (+
                 (+
                  (-
                   (/
                    (-
                     (- a)
                     (fma
                      -15.234687407
                      (+ 457.9610022158428 t)
                      1112.0901850848957))
                    z))
                  t)
                 457.9610022158428)
                z))
              36.52704169880642)
             z))
           3.13060547623)
          x)))
   (if (<= z -12.5)
     t_1
     (if (<= z 0.004)
       (fma
        y
        (/
         (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b)
         (fma 11.9400905721 z 0.607771387771))
        x)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (-((-(((-((-a - fma(-15.234687407, (457.9610022158428 + t), 1112.0901850848957)) / z) + t) + 457.9610022158428) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	double tmp;
	if (z <= -12.5) {
		tmp = t_1;
	} else if (z <= 0.004) {
		tmp = fma(y, (fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / fma(11.9400905721, z, 0.607771387771)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-a) - fma(-15.234687407, Float64(457.9610022158428 + t), 1112.0901850848957)) / z)) + t) + 457.9610022158428) / z)) + 36.52704169880642) / z)) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -12.5)
		tmp = t_1;
	elseif (z <= 0.004)
		tmp = fma(y, Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / fma(11.9400905721, z, 0.607771387771)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-N[(N[((-N[(N[(N[((-N[(N[((-a) - N[(-15.234687407 * N[(457.9610022158428 + t), $MachinePrecision] + 1112.0901850848957), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision] + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -12.5], t$95$1, If[LessEqual[z, 0.004], N[(y * N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -12.5 or 0.0040000000000000001 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + \left(t + -1 \cdot \frac{-1 \cdot a - \left(\frac{1112090185084895700201045470302189}{1000000000000000000000000000000} + \frac{-15234687407}{1000000000} \cdot \left(\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t\right)\right)}{z}\right)}{z}}{z}}, x\right) \]

   9. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(-15.234687407, 457.9610022158428 + t, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]

   if -12.5 < z < 0.0040000000000000001

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right)}{\color{blue}{\frac{607771387771}{1000000000000} + \frac{119400905721}{10000000000} \cdot z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 54.4

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

   9. Applied rewrites 54.4%

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

Alternative 3: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), x\right)\\ \mathbf{elif}\;z \leq 60:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -12.5)
   (fma
    y
    (+
     3.13060547623
     (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
    x)
   (if (<= z 60.0)
     (fma
      y
      (/
       (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b)
       (fma 11.9400905721 z 0.607771387771))
      x)
     (fma
      y
      (+
       (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
       3.13060547623)
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -12.5) {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), x);
	} else if (z <= 60.0) {
		tmp = fma(y, (fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / fma(11.9400905721, z, 0.607771387771)), x);
	} else {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -12.5)
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), x);
	elseif (z <= 60.0)
		tmp = fma(y, Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / fma(11.9400905721, z, 0.607771387771)), x);
	else
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -12.5], N[(y * N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 60.0], N[(y * N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -12.5

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \color{blue}{\frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}\right), x\right) \]

      4. div-add-rev N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000} \cdot 1}{\color{blue}{z}}\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000}}{z}\right), x\right) \]

      11. lower-/.f64 55.8

          \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{\color{blue}{z}}\right), x\right) \]

   9. Applied rewrites 55.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)}, x\right) \]

   if -12.5 < z < 60

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{313060547623}{100000000000}, z, \frac{55833770631}{5000000000}\right), z, t\right), z, a\right), z, b\right)}{\color{blue}{\frac{607771387771}{1000000000000} + \frac{119400905721}{10000000000} \cdot z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 54.4

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

   9. Applied rewrites 54.4%

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

   if 60 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -62000:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), x\right)\\ \mathbf{elif}\;z \leq 48:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -62000.0)
   (fma
    y
    (+
     3.13060547623
     (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
    x)
   (if (<= z 48.0)
     (fma
      y
      (/
       (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b)
       0.607771387771)
      x)
     (fma
      y
      (+
       (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
       3.13060547623)
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -62000.0) {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), x);
	} else if (z <= 48.0) {
		tmp = fma(y, (fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / 0.607771387771), x);
	} else {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -62000.0)
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), x);
	elseif (z <= 48.0)
		tmp = fma(y, Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / 0.607771387771), x);
	else
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -62000.0], N[(y * N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 48.0], N[(y * N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(y * N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -62000

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \color{blue}{\frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}\right), x\right) \]

      4. div-add-rev N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000} \cdot 1}{\color{blue}{z}}\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000}}{z}\right), x\right) \]

      11. lower-/.f64 55.8

          \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{\color{blue}{z}}\right), x\right) \]

   9. Applied rewrites 55.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)}, x\right) \]

   if -62000 < z < 48

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 55.1%

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

   if 48 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -240:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), x\right)\\ \mathbf{elif}\;z \leq 0.012:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998 \cdot a, y, -32.324150453290734 \cdot \left(b \cdot y\right)\right), z, \left(b \cdot y\right) \cdot 1.6453555072203998\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -240.0)
   (fma
    y
    (+
     3.13060547623
     (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
    x)
   (if (<= z 0.012)
     (+
      (fma
       (fma (* 1.6453555072203998 a) y (* -32.324150453290734 (* b y)))
       z
       (* (* b y) 1.6453555072203998))
      x)
     (fma
      y
      (+
       (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
       3.13060547623)
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -240.0) {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), x);
	} else if (z <= 0.012) {
		tmp = fma(fma((1.6453555072203998 * a), y, (-32.324150453290734 * (b * y))), z, ((b * y) * 1.6453555072203998)) + x;
	} else {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -240.0)
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), x);
	elseif (z <= 0.012)
		tmp = Float64(fma(fma(Float64(1.6453555072203998 * a), y, Float64(-32.324150453290734 * Float64(b * y))), z, Float64(Float64(b * y) * 1.6453555072203998)) + x);
	else
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -240.0], N[(y * N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 0.012], N[(N[(N[(N[(1.6453555072203998 * a), $MachinePrecision] * y + N[(-32.324150453290734 * N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(N[(b * y), $MachinePrecision] * 1.6453555072203998), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -240

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \color{blue}{\frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}\right), x\right) \]

      4. div-add-rev N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000} \cdot 1}{\color{blue}{z}}\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000}}{z}\right), x\right) \]

      11. lower-/.f64 55.8

          \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{\color{blue}{z}}\right), x\right) \]

   9. Applied rewrites 55.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)}, x\right) \]

   if -240 < z < 0.012

   1. Initial program 58.6%

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

   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. lower-+.f64 N/A

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

   4. Applied rewrites 52.3%

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

   if 0.012 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -19:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), x\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{y \cdot b}{\left(11.9400905721 - -31.4690115749 \cdot z\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -19.0)
   (fma
    y
    (+
     3.13060547623
     (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
    x)
   (if (<= z 6.8e-13)
     (+
      x
      (/
       (* y b)
       (+ (* (- 11.9400905721 (* -31.4690115749 z)) z) 0.607771387771)))
     (fma
      y
      (+
       (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
       3.13060547623)
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -19.0) {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), x);
	} else if (z <= 6.8e-13) {
		tmp = x + ((y * b) / (((11.9400905721 - (-31.4690115749 * z)) * z) + 0.607771387771));
	} else {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -19.0)
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), x);
	elseif (z <= 6.8e-13)
		tmp = Float64(x + Float64(Float64(y * b) / Float64(Float64(Float64(11.9400905721 - Float64(-31.4690115749 * z)) * z) + 0.607771387771)));
	else
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -19.0], N[(y * N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.8e-13], N[(x + N[(N[(y * b), $MachinePrecision] / N[(N[(N[(11.9400905721 - N[(-31.4690115749 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -19

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   8. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \color{blue}{\left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)}, x\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right) - \color{blue}{\frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}\right), x\right) \]

      4. div-add-rev N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \color{blue}{\frac{3652704169880641883561}{100000000000000000000}} \cdot \frac{1}{z}\right), x\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000} \cdot 1}{\color{blue}{z}}\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z \cdot z} - \frac{\frac{3652704169880641883561}{100000000000000000000}}{z}\right), x\right) \]

      11. lower-/.f64 55.8

          \[\leadsto \mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{\color{blue}{z}}\right), x\right) \]

   9. Applied rewrites 55.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)}, x\right) \]

   if -19 < z < 6.80000000000000031e-13

   1. Initial program 58.6%

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

   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 64.8%

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

   5. Taylor expanded in z around 0

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

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

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot \color{blue}{z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. metadata-eval 64.0

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

   7. Applied rewrites 64.0%

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

   if 6.80000000000000031e-13 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{y \cdot b}{\left(11.9400905721 - -31.4690115749 \cdot z\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          y
          (+
           (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
           3.13060547623)
          x)))
   (if (<= z -19.0)
     t_1
     (if (<= z 6.8e-13)
       (+
        x
        (/
         (* y b)
         (+ (* (- 11.9400905721 (* -31.4690115749 z)) 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, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	double tmp;
	if (z <= -19.0) {
		tmp = t_1;
	} else if (z <= 6.8e-13) {
		tmp = x + ((y * b) / (((11.9400905721 - (-31.4690115749 * z)) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -19.0)
		tmp = t_1;
	elseif (z <= 6.8e-13)
		tmp = Float64(x + Float64(Float64(y * b) / Float64(Float64(Float64(11.9400905721 - Float64(-31.4690115749 * z)) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -19.0], t$95$1, If[LessEqual[z, 6.8e-13], N[(x + N[(N[(y * b), $MachinePrecision] / N[(N[(N[(11.9400905721 - N[(-31.4690115749 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -19 or 6.80000000000000031e-13 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]

   if -19 < z < 6.80000000000000031e-13

   1. Initial program 58.6%

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

   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 64.8%

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

   5. Taylor expanded in z around 0

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

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

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot \color{blue}{z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. metadata-eval 64.0

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

   7. Applied rewrites 64.0%

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

Alternative 8: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{y \cdot b}{\left(11.9400905721 - -31.4690115749 \cdot z\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (+ (- (- (/ t (* z z)))) 3.13060547623) x)))
   (if (<= z -19.0)
     t_1
     (if (<= z 6.8e-13)
       (+
        x
        (/
         (* y b)
         (+ (* (- 11.9400905721 (* -31.4690115749 z)) 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, (-(-(t / (z * z))) + 3.13060547623), x);
	double tmp;
	if (z <= -19.0) {
		tmp = t_1;
	} else if (z <= 6.8e-13) {
		tmp = x + ((y * b) / (((11.9400905721 - (-31.4690115749 * z)) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(-Float64(t / Float64(z * z)))) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -19.0)
		tmp = t_1;
	elseif (z <= 6.8e-13)
		tmp = Float64(x + Float64(Float64(y * b) / Float64(Float64(Float64(11.9400905721 - Float64(-31.4690115749 * z)) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-(-N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision])) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -19.0], t$95$1, If[LessEqual[z, 6.8e-13], N[(x + N[(N[(y * b), $MachinePrecision] / N[(N[(N[(11.9400905721 - N[(-31.4690115749 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -19 or 6.80000000000000031e-13 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{fma}\left(y, \left(--1 \cdot \frac{t}{{z}^{2}}\right) + \frac{313060547623}{100000000000}, x\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(\mathsf{neg}\left(\frac{t}{{z}^{2}}\right)\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{{z}^{2}}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{{z}^{2}}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       5. lift-*.f64 56.7

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + 3.13060547623, x\right) \]

   12. Applied rewrites 56.7%

       \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + 3.13060547623, x\right) \]

   if -19 < z < 6.80000000000000031e-13

   1. Initial program 58.6%

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

   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 64.8%

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

   5. Taylor expanded in z around 0

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

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

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \color{blue}{\left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} - \left(\mathsf{neg}\left(\frac{314690115749}{10000000000}\right)\right) \cdot \color{blue}{z}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. metadata-eval 64.0

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

   7. Applied rewrites 64.0%

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

Alternative 9: 86.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -19:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (+ (- (- (/ t (* z z)))) 3.13060547623) x)))
   (if (<= z -19.0)
     t_1
     (if (<= z 6.8e-13) (fma (* b y) 1.6453555072203998 x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (-(-(t / (z * z))) + 3.13060547623), x);
	double tmp;
	if (z <= -19.0) {
		tmp = t_1;
	} else if (z <= 6.8e-13) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(-Float64(t / Float64(z * z)))) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -19.0)
		tmp = t_1;
	elseif (z <= 6.8e-13)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-(-N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision])) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -19.0], t$95$1, If[LessEqual[z, 6.8e-13], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -19 or 6.80000000000000031e-13 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{fma}\left(y, \left(--1 \cdot \frac{t}{{z}^{2}}\right) + \frac{313060547623}{100000000000}, x\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(\mathsf{neg}\left(\frac{t}{{z}^{2}}\right)\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       2. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{{z}^{2}}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       3. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{{z}^{2}}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

       5. lift-*.f64 56.7

          \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + 3.13060547623, x\right) \]

   12. Applied rewrites 56.7%

       \[\leadsto \mathsf{fma}\left(y, \left(-\left(-\frac{t}{z \cdot z}\right)\right) + 3.13060547623, x\right) \]

   if -19 < z < 6.80000000000000031e-13

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 60.8

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

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -60000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -60000.0)
     t_1
     (if (<= z 6.8e-13)
       (fma (* b y) 1.6453555072203998 x)
       (if (<= z 1.7e+50) (fma y (/ t (* z z)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (36.52704169880642 / z)), x);
	double tmp;
	if (z <= -60000.0) {
		tmp = t_1;
	} else if (z <= 6.8e-13) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else if (z <= 1.7e+50) {
		tmp = fma(y, (t / (z * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(36.52704169880642 / z)), x)
	tmp = 0.0
	if (z <= -60000.0)
		tmp = t_1;
	elseif (z <= 6.8e-13)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	elseif (z <= 1.7e+50)
		tmp = fma(y, Float64(t / Float64(z * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -60000.0], t$95$1, If[LessEqual[z, 6.8e-13], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], If[LessEqual[z, 1.7e+50], N[(y * N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6e4 or 1.6999999999999999e50 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - \frac{\frac{3652704169880641883561}{100000000000000000000}}{z}, x\right) \]

      4. lower-/.f64 59.6

         \[\leadsto \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{\color{blue}{z}}, x\right) \]

   9. Applied rewrites 59.6%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \frac{36.52704169880642}{z}}, x\right) \]

   if -6e4 < z < 6.80000000000000031e-13

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 60.8

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

   4. Applied rewrites 60.8%

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

   if 6.80000000000000031e-13 < z < 1.6999999999999999e50

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z} + \color{blue}{\frac{313060547623}{100000000000}}, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\mathsf{neg}\left(\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right)\right) + \frac{313060547623}{100000000000}, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z} + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(\mathsf{neg}\left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right)\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}\right) + \frac{3652704169880641883561}{100000000000000000000}}{z}\right) + \frac{313060547623}{100000000000}, x\right) \]

      11. lower-+.f64 55.9

          \[\leadsto \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right) \]

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623}, x\right) \]

   10. Taylor expanded in t around inf

       \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{t}{{z}^{\color{blue}{2}}}, x\right) \]

       2. pow2 N/A

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

       3. lift-*.f64 36.9

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

   12. Applied rewrites 36.9%

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

Alternative 11: 83.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -60000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -60000.0)
     t_1
     (if (<= z 3.8e-9) (fma (* b y) 1.6453555072203998 x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (36.52704169880642 / z)), x);
	double tmp;
	if (z <= -60000.0) {
		tmp = t_1;
	} else if (z <= 3.8e-9) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(36.52704169880642 / z)), x)
	tmp = 0.0
	if (z <= -60000.0)
		tmp = t_1;
	elseif (z <= 3.8e-9)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -60000.0], t$95$1, If[LessEqual[z, 3.8e-9], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e4 or 3.80000000000000011e-9 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around inf

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

      1. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{\left(2 + \color{blue}{2}\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{{z}^{2} \cdot \color{blue}{{z}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot {\color{blue}{z}}^{2}} \]

      6. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(z \cdot z\right) \cdot \left(z \cdot \color{blue}{z}\right)} \]

      7. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   6. Applied rewrites 21.6%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - \frac{\frac{3652704169880641883561}{100000000000000000000}}{z}, x\right) \]

      4. lower-/.f64 59.6

         \[\leadsto \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{\color{blue}{z}}, x\right) \]

   9. Applied rewrites 59.6%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \frac{36.52704169880642}{z}}, x\right) \]

   if -6e4 < z < 3.80000000000000011e-9

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 60.8

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

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 83.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -17000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 0.004)
     (fma (* b y) 1.6453555072203998 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 <= -17000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 0.004) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 0.004)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -17000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 0.004], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e7 or 0.0040000000000000001 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 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 63.5

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

   4. Applied rewrites 63.5%

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

   if -1.7e7 < z < 0.0040000000000000001

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 60.8

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

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 83.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(1.6453555072203998 \cdot b, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -17000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 0.004)
     (fma (* 1.6453555072203998 b) y 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 <= -17000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 0.004) {
		tmp = fma((1.6453555072203998 * b), y, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -17000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 0.004)
		tmp = fma(Float64(1.6453555072203998 * b), y, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -17000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 0.004], N[(N[(1.6453555072203998 * b), $MachinePrecision] * y + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e7 or 0.0040000000000000001 < z

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 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 63.5

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

   4. Applied rewrites 63.5%

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

   if -1.7e7 < z < 0.0040000000000000001

   1. Initial program 58.6%

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

   2. Taylor expanded in z around 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 63.5

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.6453555072203998 \cdot b, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 63.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3.13060547623, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma 3.13060547623 y x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(3.13060547623, y, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(3.13060547623, y, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(3.13060547623 * y + x), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

2. Taylor expanded in z around 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 63.5

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

4. Applied rewrites 63.5%

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

Alternative 15: 22.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ 3.13060547623 \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 3.13060547623 y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 3.13060547623 * y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 = 3.13060547623d0 * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 3.13060547623 * y;
}

Python

def code(x, y, z, t, a, b):
	return 3.13060547623 * y

Julia

function code(x, y, z, t, a, b)
	return Float64(3.13060547623 * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 3.13060547623 * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(3.13060547623 * y), $MachinePrecision]

Derivation

1. Initial program 58.6%

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

2. Taylor expanded in z around 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 63.5

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

4. Applied rewrites 63.5%

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

5. Taylor expanded in x around 0

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

   1. lift-*.f64 22.3

      \[\leadsto 3.13060547623 \cdot y \]

7. Applied rewrites 22.3%

   \[\leadsto 3.13060547623 \cdot \color{blue}{y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))