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

Percentage Accurate: 57.7% → 98.4%

Time: 7.2s

Alternatives: 22

Speedup: 3.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(457.9610022158428 + t, -15.234687407, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.35e+17)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 8.8e+36)
     (fma
      (/
       (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b)
       (fma
        (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
        z
        0.607771387771))
      y
      x)
     (+
      x
      (*
       (+
        (-
         (/
          (+
           (-
            (/
             (+
              (+
               (-
                (/
                 (-
                  (- a)
                  (fma
                   (+ 457.9610022158428 t)
                   -15.234687407
                   1112.0901850848957))
                 z))
               t)
              457.9610022158428)
             z))
           36.52704169880642)
          z))
        3.13060547623)
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.35e+17) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 8.8e+36) {
		tmp = fma((fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)), y, x);
	} else {
		tmp = x + ((-((-(((-((-a - fma((457.9610022158428 + t), -15.234687407, 1112.0901850848957)) / z) + t) + 457.9610022158428) / z) + 36.52704169880642) / z) + 3.13060547623) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.35e+17)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 8.8e+36)
		tmp = fma(Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) / fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)), y, x);
	else
		tmp = Float64(x + Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-a) - fma(Float64(457.9610022158428 + t), -15.234687407, 1112.0901850848957)) / z)) + t) + 457.9610022158428) / z)) + 36.52704169880642) / z)) + 3.13060547623) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.35e+17], 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, 8.8e+36], N[(N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(N[((-N[(N[((-N[(N[(N[((-N[(N[((-a) - N[(N[(457.9610022158428 + t), $MachinePrecision] * -15.234687407 + 1112.0901850848957), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision] + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e17

   1. Initial program 12.5%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.9

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

   7. Applied rewrites 46.9%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.5%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 95.6

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

   12. Applied rewrites 95.6%

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

   if -1.35e17 < z < 8.80000000000000002e36

   1. Initial program 98.8%

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

   3. Applied rewrites 99.5%

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

   if 8.80000000000000002e36 < z

   1. Initial program 8.1%

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

   3. Applied rewrites 12.4%

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

   4. Taylor expanded in z around -inf

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

   6. Applied rewrites 98.8%

      \[\leadsto x + \color{blue}{\left(\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(457.9610022158428 + t, -15.234687407, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623\right)} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+34}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(11.1667541262, z, t\right), z, a\right), z, b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(457.9610022158428 + t, -15.234687407, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.2e+16)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 1.9e+34)
     (+
      x
      (/
       (* y (fma (fma (fma 11.1667541262 z t) z a) z b))
       (+
        (*
         (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
         z)
        0.607771387771)))
     (+
      x
      (*
       (+
        (-
         (/
          (+
           (-
            (/
             (+
              (+
               (-
                (/
                 (-
                  (- a)
                  (fma
                   (+ 457.9610022158428 t)
                   -15.234687407
                   1112.0901850848957))
                 z))
               t)
              457.9610022158428)
             z))
           36.52704169880642)
          z))
        3.13060547623)
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.2e+16) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 1.9e+34) {
		tmp = x + ((y * fma(fma(fma(11.1667541262, z, t), z, a), z, b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = x + ((-((-(((-((-a - fma((457.9610022158428 + t), -15.234687407, 1112.0901850848957)) / z) + t) + 457.9610022158428) / z) + 36.52704169880642) / z) + 3.13060547623) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.2e+16)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 1.9e+34)
		tmp = Float64(x + Float64(Float64(y * fma(fma(fma(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 = Float64(x + Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-a) - fma(Float64(457.9610022158428 + t), -15.234687407, 1112.0901850848957)) / z)) + t) + 457.9610022158428) / z)) + 36.52704169880642) / z)) + 3.13060547623) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.2e+16], 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, 1.9e+34], N[(x + N[(N[(y * N[(N[(N[(11.1667541262 * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + 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], N[(x + N[(N[((-N[(N[((-N[(N[(N[((-N[(N[((-a) - N[(N[(457.9610022158428 + t), $MachinePrecision] * -15.234687407 + 1112.0901850848957), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision] + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e16

   1. Initial program 12.6%

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

   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 47.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.9

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

   7. Applied rewrites 46.9%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.5%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 95.6

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

   12. Applied rewrites 95.6%

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

   if -5.2e16 < z < 1.9000000000000001e34

   1. Initial program 98.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.2

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

   4. Applied rewrites 98.2%

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

   if 1.9000000000000001e34 < z

   1. Initial program 8.8%

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

   3. Applied rewrites 13.1%

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

   4. Taylor expanded in z around -inf

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

   6. Applied rewrites 98.7%

      \[\leadsto x + \color{blue}{\left(\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(457.9610022158428 + t, -15.234687407, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623\right)} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(457.9610022158428 + t, -15.234687407, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.5e+16)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 2.5e+34)
     (+
      x
      (*
       (/
        (fma (fma t z a) z b)
        (fma
         (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
         z
         0.607771387771))
       y))
     (+
      x
      (*
       (+
        (-
         (/
          (+
           (-
            (/
             (+
              (+
               (-
                (/
                 (-
                  (- a)
                  (fma
                   (+ 457.9610022158428 t)
                   -15.234687407
                   1112.0901850848957))
                 z))
               t)
              457.9610022158428)
             z))
           36.52704169880642)
          z))
        3.13060547623)
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.5e+16) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 2.5e+34) {
		tmp = x + ((fma(fma(t, z, a), z, b) / fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)) * y);
	} else {
		tmp = x + ((-((-(((-((-a - fma((457.9610022158428 + t), -15.234687407, 1112.0901850848957)) / z) + t) + 457.9610022158428) / z) + 36.52704169880642) / z) + 3.13060547623) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.5e+16)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 2.5e+34)
		tmp = Float64(x + Float64(Float64(fma(fma(t, z, a), z, b) / fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)) * y));
	else
		tmp = Float64(x + Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(Float64(-a) - fma(Float64(457.9610022158428 + t), -15.234687407, 1112.0901850848957)) / z)) + t) + 457.9610022158428) / z)) + 36.52704169880642) / z)) + 3.13060547623) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.5e+16], 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, 2.5e+34], N[(x + N[(N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[((-N[(N[((-N[(N[(N[((-N[(N[((-a) - N[(N[(457.9610022158428 + t), $MachinePrecision] * -15.234687407 + 1112.0901850848957), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision] + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e16

   1. Initial program 12.6%

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

   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 47.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.9

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

   7. Applied rewrites 46.9%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.5%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 95.6

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

   12. Applied rewrites 95.6%

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

   if -5.5e16 < z < 2.4999999999999999e34

   1. Initial program 98.9%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 98.3%

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

   if 2.4999999999999999e34 < z

   1. Initial program 8.8%

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

   3. Applied rewrites 13.1%

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

   4. Taylor expanded in z around -inf

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

   6. Applied rewrites 98.7%

      \[\leadsto x + \color{blue}{\left(\left(-\frac{\left(-\frac{\left(\left(-\frac{\left(-a\right) - \mathsf{fma}\left(457.9610022158428 + t, -15.234687407, 1112.0901850848957\right)}{z}\right) + t\right) + 457.9610022158428}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623\right)} \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.5e+16)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 1.22e+37)
     (+
      x
      (*
       (/
        (fma (fma t z a) z b)
        (fma
         (fma (fma (+ 15.234687407 z) z 31.4690115749) z 11.9400905721)
         z
         0.607771387771))
       y))
     (+
      x
      (*
       (+
        3.13060547623
        (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.5e+16) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 1.22e+37) {
		tmp = x + ((fma(fma(t, z, a), z, b) / fma(fma(fma((15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)) * y);
	} else {
		tmp = x + ((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.5e+16)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 1.22e+37)
		tmp = Float64(x + Float64(Float64(fma(fma(t, z, a), z, b) / fma(fma(fma(Float64(15.234687407 + z), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)) * y));
	else
		tmp = Float64(x + Float64(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.5e+16], 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, 1.22e+37], N[(x + N[(N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(N[(N[(N[(15.234687407 + z), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e16

   1. Initial program 12.6%

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

   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 47.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.9

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

   7. Applied rewrites 46.9%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.5%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 95.6

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

   12. Applied rewrites 95.6%

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

   if -5.5e16 < z < 1.22e37

   1. Initial program 98.8%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 98.2%

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

   if 1.22e37 < z

   1. Initial program 8.1%

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

   3. Applied rewrites 12.4%

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

   4. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.4

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

   6. Applied rewrites 97.4%

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

Alternative 5: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -900000000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+33}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(11.1667541262, z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(31.4690115749, z, 11.9400905721\right), z, 0.607771387771\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -900000000.0)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 7.2e+33)
     (+
      x
      (/
       (* y (fma (fma (fma 11.1667541262 z t) z a) z b))
       (fma (fma 31.4690115749 z 11.9400905721) z 0.607771387771)))
     (+
      x
      (*
       (+
        3.13060547623
        (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -900000000.0) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 7.2e+33) {
		tmp = x + ((y * fma(fma(fma(11.1667541262, z, t), z, a), z, b)) / fma(fma(31.4690115749, z, 11.9400905721), z, 0.607771387771));
	} else {
		tmp = x + ((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -900000000.0)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 7.2e+33)
		tmp = Float64(x + Float64(Float64(y * fma(fma(fma(11.1667541262, z, t), z, a), z, b)) / fma(fma(31.4690115749, z, 11.9400905721), z, 0.607771387771)));
	else
		tmp = Float64(x + Float64(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -900000000.0], 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, 7.2e+33], N[(x + N[(N[(y * N[(N[(N[(11.1667541262 * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(31.4690115749 * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9e8

   1. Initial program 14.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.8

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

   7. Applied rewrites 46.8%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.3%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 94.8

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

   12. Applied rewrites 94.8%

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

   if -9e8 < z < 7.2000000000000005e33

   1. Initial program 98.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{55833770631}{5000000000}, z, t\right), z, a\right), z, b\right)}{\color{blue}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + \frac{314690115749}{10000000000} \cdot z\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 7.2000000000000005e33 < z

   1. Initial program 8.9%

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

   3. Applied rewrites 13.2%

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

   4. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.2

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

   6. Applied rewrites 97.2%

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

Alternative 6: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(11.1667541262, z, t\right), z, a\right), z, b\right)}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -13.0)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 7.2e+33)
     (fma
      y
      (/
       (fma (fma (fma 11.1667541262 z t) z a) z b)
       (fma 11.9400905721 z 0.607771387771))
      x)
     (+
      x
      (*
       (+
        3.13060547623
        (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -13.0) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 7.2e+33) {
		tmp = fma(y, (fma(fma(fma(11.1667541262, z, t), z, a), z, b) / fma(11.9400905721, z, 0.607771387771)), x);
	} else {
		tmp = x + ((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -13.0)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 7.2e+33)
		tmp = fma(y, Float64(fma(fma(fma(11.1667541262, z, t), z, a), z, b) / fma(11.9400905721, z, 0.607771387771)), x);
	else
		tmp = Float64(x + Float64(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -13.0], 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, 7.2e+33], N[(y * N[(N[(N[(N[(11.1667541262 * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 16.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.5

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

   7. Applied rewrites 46.5%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.0%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 93.6

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

   12. Applied rewrites 93.6%

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

   if -13 < z < 7.2000000000000005e33

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.6

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

   7. Applied rewrites 96.6%

      \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(11.1667541262, z, t\right), z, a\right), z, b\right)}{\color{blue}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 96.6%

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

   if 7.2000000000000005e33 < z

   1. Initial program 8.9%

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

   3. Applied rewrites 13.2%

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

   4. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.2

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

   6. Applied rewrites 97.2%

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

Alternative 7: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+33}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\left(\frac{0.607771387771}{z} + 11.9400905721\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -13.0)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 7.2e+33)
     (+
      x
      (/
       (* y (fma (fma t z a) z b))
       (* (+ (/ 0.607771387771 z) 11.9400905721) z)))
     (+
      x
      (*
       (+
        3.13060547623
        (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -13.0) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 7.2e+33) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / (((0.607771387771 / z) + 11.9400905721) * z));
	} else {
		tmp = x + ((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -13.0)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 7.2e+33)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / Float64(Float64(Float64(0.607771387771 / z) + 11.9400905721) * z)));
	else
		tmp = Float64(x + Float64(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -13.0], 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, 7.2e+33], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.607771387771 / z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 16.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.5

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

   7. Applied rewrites 46.5%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.0%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 93.6

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

   12. Applied rewrites 93.6%

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

   if -13 < z < 7.2000000000000005e33

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.6

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in z around inf

      \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{55833770631}{5000000000}, z, t\right), z, a\right), z, b\right)}{z \cdot \color{blue}{\left(\frac{119400905721}{10000000000} + \frac{607771387771}{1000000000000} \cdot \frac{1}{z}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 96.5

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

   10. Applied rewrites 96.5%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 96.2%

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

   if 7.2000000000000005e33 < z

   1. Initial program 8.9%

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

   3. Applied rewrites 13.2%

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

   4. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.2

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

   6. Applied rewrites 97.2%

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

Alternative 8: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.062:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+33}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(11.1667541262, z, t\right), z, a\right), z, b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.062)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 7.2e+33)
     (+ x (/ (* y (fma (fma (fma 11.1667541262 z t) z a) z b)) 0.607771387771))
     (+
      x
      (*
       (+
        3.13060547623
        (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.062) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 7.2e+33) {
		tmp = x + ((y * fma(fma(fma(11.1667541262, z, t), z, a), z, b)) / 0.607771387771);
	} else {
		tmp = x + ((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.062)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 7.2e+33)
		tmp = Float64(x + Float64(Float64(y * fma(fma(fma(11.1667541262, z, t), z, a), z, b)) / 0.607771387771));
	else
		tmp = Float64(x + Float64(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.062], 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, 7.2e+33], N[(x + N[(N[(y * N[(N[(N[(11.1667541262 * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.062

   1. Initial program 17.3%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.5

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

   7. Applied rewrites 46.5%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.1%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 93.3

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

   12. Applied rewrites 93.3%

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

   if -0.062 < z < 7.2000000000000005e33

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 96.0%

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

   if 7.2000000000000005e33 < z

   1. Initial program 8.9%

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

   3. Applied rewrites 13.2%

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

   4. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.2

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

   6. Applied rewrites 97.2%

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

Alternative 9: 92.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;x + \mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.2e-16)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 1.1e+31)
     (+
      x
      (*
       (fma
        (fma 1.6453555072203998 a (* -32.324150453290734 b))
        z
        (* 1.6453555072203998 b))
       y))
     (+
      x
      (*
       (+
        3.13060547623
        (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
       y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.2e-16) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 1.1e+31) {
		tmp = x + (fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)) * y);
	} else {
		tmp = x + ((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.2e-16)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 1.1e+31)
		tmp = Float64(x + Float64(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)) * y));
	else
		tmp = Float64(x + Float64(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.2e-16], 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, 1.1e+31], N[(x + N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.20000000000000023e-16

   1. Initial program 21.1%

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

   2. Taylor expanded in z around 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 48.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.9

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

   7. Applied rewrites 46.9%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.4%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 91.0

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

   12. Applied rewrites 91.0%

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

   if -3.20000000000000023e-16 < z < 1.10000000000000005e31

   1. Initial program 99.1%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.1

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

   6. Applied rewrites 91.1%

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

   if 1.10000000000000005e31 < z

   1. Initial program 9.4%

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

   3. Applied rewrites 13.8%

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

   4. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 96.9

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

   6. Applied rewrites 96.9%

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

Alternative 10: 92.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;x + \mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.2e-16)
   (fma
    y
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    x)
   (if (<= z 1.1e+31)
     (+
      x
      (*
       (fma
        (fma 1.6453555072203998 a (* -32.324150453290734 b))
        z
        (* 1.6453555072203998 b))
       y))
     (fma
      y
      (+
       3.13060547623
       (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.2e-16) {
		tmp = fma(y, (-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), x);
	} else if (z <= 1.1e+31) {
		tmp = x + (fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)) * y);
	} else {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.2e-16)
		tmp = fma(y, Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), x);
	elseif (z <= 1.1e+31)
		tmp = Float64(x + Float64(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)) * y));
	else
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.2e-16], 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, 1.1e+31], N[(x + N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.20000000000000023e-16

   1. Initial program 21.1%

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

   2. Taylor expanded in z around 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 48.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 46.9

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

   7. Applied rewrites 46.9%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 48.4%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 91.0

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

   12. Applied rewrites 91.0%

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

   if -3.20000000000000023e-16 < z < 1.10000000000000005e31

   1. Initial program 99.1%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.1

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

   6. Applied rewrites 91.1%

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

   if 1.10000000000000005e31 < z

   1. Initial program 9.4%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 49.3

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

   7. Applied rewrites 49.3%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 51.5%

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

   10. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{313060547623}{100000000000} + \left(\frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000}}{{z}^{2}} + \frac{t}{{z}^{2}}\right)\right) - \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}}, x\right) \]
   11. 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. unpow2 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. lower-*.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 96.9

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

   12. Applied rewrites 96.9%

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

Alternative 11: 92.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;x + \mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -3.2e-16)
     t_1
     (if (<= z 1.1e+31)
       (+
        x
        (*
         (fma
          (fma 1.6453555072203998 a (* -32.324150453290734 b))
          z
          (* 1.6453555072203998 b))
         y))
       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 <= -3.2e-16) {
		tmp = t_1;
	} else if (z <= 1.1e+31) {
		tmp = x + (fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)) * y);
	} 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 <= -3.2e-16)
		tmp = t_1;
	elseif (z <= 1.1e+31)
		tmp = Float64(x + Float64(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)) * y));
	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, -3.2e-16], t$95$1, If[LessEqual[z, 1.1e+31], N[(x + N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000023e-16 or 1.10000000000000005e31 < z

   1. Initial program 15.8%

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

   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 49.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right)\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(z \cdot \left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z\right) \cdot z + \frac{119400905721}{10000000000}, z, \frac{607771387771}{1000000000000}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{314690115749}{10000000000} + \frac{15234687407}{1000000000} \cdot z, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{15234687407}{1000000000} \cdot z + \frac{314690115749}{10000000000}, z, \frac{119400905721}{10000000000}\right), z, \frac{607771387771}{1000000000000}\right)} \]

      8. lower-fma.f64 48.0

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

   7. Applied rewrites 48.0%

      \[\leadsto x + \frac{y \cdot b}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

   9. Applied rewrites 49.8%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{313060547623}{100000000000} + -1 \cdot \frac{\frac{3652704169880641883561}{100000000000000000000} + -1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{z}}{z}}, x\right) \]
   11. 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 93.6

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

   12. Applied rewrites 93.6%

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

   if -3.20000000000000023e-16 < z < 1.10000000000000005e31

   1. Initial program 99.1%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.1

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

   6. Applied rewrites 91.1%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right)} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 90.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -0.06:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;x + \mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -0.06)
     t_1
     (if (<= z 1.6e+32)
       (+
        x
        (*
         (fma
          (fma 1.6453555072203998 a (* -32.324150453290734 b))
          z
          (* 1.6453555072203998 b))
         y))
       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 <= -0.06) {
		tmp = t_1;
	} else if (z <= 1.6e+32) {
		tmp = x + (fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)) * y);
	} 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 <= -0.06)
		tmp = t_1;
	elseif (z <= 1.6e+32)
		tmp = Float64(x + Float64(fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)) * y));
	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, -0.06], t$95$1, If[LessEqual[z, 1.6e+32], N[(x + N[(N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.059999999999999998 or 1.5999999999999999e32 < z

   1. Initial program 13.6%

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

   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 48.9%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 16.2%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 89.5

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

   10. Applied rewrites 89.5%

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

   if -0.059999999999999998 < z < 1.5999999999999999e32

   1. Initial program 99.1%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.6

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

   6. Applied rewrites 90.6%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right)} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -0.06:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), z, 1.6453555072203998 \cdot b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -0.06)
     t_1
     (if (<= z 1.6e+32)
       (fma
        y
        (fma
         (fma 1.6453555072203998 a (* -32.324150453290734 b))
         z
         (* 1.6453555072203998 b))
        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 <= -0.06) {
		tmp = t_1;
	} else if (z <= 1.6e+32) {
		tmp = fma(y, fma(fma(1.6453555072203998, a, (-32.324150453290734 * b)), z, (1.6453555072203998 * b)), 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 <= -0.06)
		tmp = t_1;
	elseif (z <= 1.6e+32)
		tmp = fma(y, fma(fma(1.6453555072203998, a, Float64(-32.324150453290734 * b)), z, Float64(1.6453555072203998 * b)), 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, -0.06], t$95$1, If[LessEqual[z, 1.6e+32], N[(y * N[(N[(1.6453555072203998 * a + N[(-32.324150453290734 * b), $MachinePrecision]), $MachinePrecision] * z + N[(1.6453555072203998 * b), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.059999999999999998 or 1.5999999999999999e32 < z

   1. Initial program 13.6%

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

   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 48.9%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 16.2%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 89.5

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

   10. Applied rewrites 89.5%

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

   if -0.059999999999999998 < z < 1.5999999999999999e32

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 90.6

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

   10. Applied rewrites 90.6%

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(1.6453555072203998, a, -32.324150453290734 \cdot b\right), \color{blue}{z}, 1.6453555072203998 \cdot b\right), x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 86.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -12200000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(t \cdot z, z, b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -12200000.0)
   (fma 3.13060547623 y x)
   (if (<= z 2.1e+37)
     (+ x (/ (* y (fma (* t z) z b)) 0.607771387771))
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -12200000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 2.1e+37) {
		tmp = x + ((y * fma((t * z), z, b)) / 0.607771387771);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -12200000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 2.1e+37)
		tmp = Float64(x + Float64(Float64(y * fma(Float64(t * z), z, b)) / 0.607771387771));
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -12200000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 2.1e+37], N[(x + N[(N[(y * N[(N[(t * z), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22e7 or 2.1000000000000001e37 < z

   1. Initial program 11.9%

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

   2. Taylor expanded in z around 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 90.5

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

   4. Applied rewrites 90.5%

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

   if -1.22e7 < z < 2.1000000000000001e37

   1. Initial program 98.9%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 83.3

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

   10. Applied rewrites 83.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 83.1%

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

Alternative 15: 82.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;x + \frac{y \cdot b}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -1.6e-62)
     t_1
     (if (<= z 90000000000.0)
       (+ x (/ (* y b) (fma 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, (3.13060547623 - (36.52704169880642 / z)), x);
	double tmp;
	if (z <= -1.6e-62) {
		tmp = t_1;
	} else if (z <= 90000000000.0) {
		tmp = x + ((y * b) / fma(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(3.13060547623 - Float64(36.52704169880642 / z)), x)
	tmp = 0.0
	if (z <= -1.6e-62)
		tmp = t_1;
	elseif (z <= 90000000000.0)
		tmp = Float64(x + Float64(Float64(y * b) / fma(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[(3.13060547623 - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.6e-62], t$95$1, If[LessEqual[z, 90000000000.0], N[(x + N[(N[(y * b), $MachinePrecision] / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000011e-62 or 9e10 < z

   1. Initial program 23.6%

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

   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 50.2%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

   if -1.60000000000000011e-62 < z < 9e10

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 16: 82.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{b}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -1.6e-62)
     t_1
     (if (<= z 90000000000.0)
       (fma y (/ 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, (3.13060547623 - (36.52704169880642 / z)), x);
	double tmp;
	if (z <= -1.6e-62) {
		tmp = t_1;
	} else if (z <= 90000000000.0) {
		tmp = fma(y, (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(3.13060547623 - Float64(36.52704169880642 / z)), x)
	tmp = 0.0
	if (z <= -1.6e-62)
		tmp = t_1;
	elseif (z <= 90000000000.0)
		tmp = fma(y, Float64(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[(3.13060547623 - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.6e-62], t$95$1, If[LessEqual[z, 90000000000.0], N[(y * N[(b / N[(11.9400905721 * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000011e-62 or 9e10 < z

   1. Initial program 23.6%

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

   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 50.2%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

   if -1.60000000000000011e-62 < z < 9e10

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + z \cdot \left(t + \frac{55833770631}{5000000000} \cdot z\right)\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 81.8%

       \[\leadsto x + \frac{y \cdot b}{\mathsf{fma}\left(11.9400905721, z, 0.607771387771\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

   12. Applied rewrites 81.8%

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

Alternative 17: 82.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{36.52704169880642}{z}, x\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ 36.52704169880642 z)) x)))
   (if (<= z -1.6e-62)
     t_1
     (if (<= z 90000000000.0) (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 <= -1.6e-62) {
		tmp = t_1;
	} else if (z <= 90000000000.0) {
		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 <= -1.6e-62)
		tmp = t_1;
	elseif (z <= 90000000000.0)
		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, -1.6e-62], t$95$1, If[LessEqual[z, 90000000000.0], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000011e-62 or 9e10 < z

   1. Initial program 23.6%

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

   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 50.2%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

   if -1.60000000000000011e-62 < z < 9e10

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 81.8

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

   4. Applied rewrites 81.8%

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

Alternative 18: 82.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.6e-62)
   (fma 3.13060547623 y x)
   (if (<= z 90000000000.0)
     (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 <= -1.6e-62) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 90000000000.0) {
		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 <= -1.6e-62)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 90000000000.0)
		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, -1.6e-62], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 90000000000.0], 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 < -1.60000000000000011e-62 or 9e10 < z

   1. Initial program 23.6%

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

   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 83.2

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

   4. Applied rewrites 83.2%

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

   if -1.60000000000000011e-62 < z < 9e10

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 81.8

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

   4. Applied rewrites 81.8%

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

Alternative 19: 82.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;\mathsf{fma}\left(y, 1.6453555072203998 \cdot b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.6e-62)
   (fma 3.13060547623 y x)
   (if (<= z 90000000000.0)
     (fma y (* 1.6453555072203998 b) x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.6e-62) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 90000000000.0) {
		tmp = fma(y, (1.6453555072203998 * b), x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.6e-62)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 90000000000.0)
		tmp = fma(y, Float64(1.6453555072203998 * b), x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.6e-62], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 90000000000.0], N[(y * N[(1.6453555072203998 * b), $MachinePrecision] + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000011e-62 or 9e10 < z

   1. Initial program 23.6%

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

   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 83.2

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

   4. Applied rewrites 83.2%

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

   if -1.60000000000000011e-62 < z < 9e10

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 95.3%

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

   8. Taylor expanded in z around 0

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

      1. lower-*.f64 81.8

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

   10. Applied rewrites 81.8%

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

Alternative 20: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \leq 2 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (+
       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)))
      2e-89)
   x
   (fma 3.13060547623 y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((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))) <= 2e-89) {
		tmp = x;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (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))) <= 2e-89)
		tmp = x;
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e-89], x, N[(3.13060547623 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))) < 2.00000000000000008e-89

   1. Initial program 94.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 41.5%

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

   if 2.00000000000000008e-89 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))))

   1. Initial program 39.3%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 74.2

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

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547623, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-215}:\\ \;\;\;\;3.13060547623 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -5.8e-92) x (if (<= x 1e-215) (* 3.13060547623 y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.8e-92) {
		tmp = x;
	} else if (x <= 1e-215) {
		tmp = 3.13060547623 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-5.8d-92)) then
        tmp = x
    else if (x <= 1d-215) then
        tmp = 3.13060547623d0 * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.8e-92) {
		tmp = x;
	} else if (x <= 1e-215) {
		tmp = 3.13060547623 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -5.8e-92:
		tmp = x
	elif x <= 1e-215:
		tmp = 3.13060547623 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -5.8e-92)
		tmp = x;
	elseif (x <= 1e-215)
		tmp = Float64(3.13060547623 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -5.8e-92)
		tmp = x;
	elseif (x <= 1e-215)
		tmp = 3.13060547623 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -5.8e-92], x, If[LessEqual[x, 1e-215], N[(3.13060547623 * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.79999999999999969e-92 or 1.00000000000000004e-215 < x

   1. Initial program 57.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 56.0%

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

   if -5.79999999999999969e-92 < x < 1.00000000000000004e-215

   1. Initial program 57.7%

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

   2. Taylor expanded in 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 45.4

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 35.7

         \[\leadsto 3.13060547623 \cdot y \]

   7. Applied rewrites 35.7%

      \[\leadsto 3.13060547623 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 45.6% accurate, 52.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 45.6%

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

Reproduce

herbie shell --seed 2025106 
(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))))