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

Percentage Accurate: 58.3% → 97.5%

Time: 6.0s

Alternatives: 12

Speedup: 8.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 58.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(3.13060547623, y, -\left(-t \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma 3.13060547623 y (- (- (* t (/ y (* z z)))))))))
   (if (<= z -2.2e+14)
     t_1
     (if (<= z 3.3e+19)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (*
           (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
           z)
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(3.13060547623, y, -(-(t * (y / (z * z)))));
	double tmp;
	if (z <= -2.2e+14) {
		tmp = t_1;
	} else if (z <= 3.3e+19) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(3.13060547623, y, Float64(-Float64(-Float64(t * Float64(y / Float64(z * z)))))))
	tmp = 0.0
	if (z <= -2.2e+14)
		tmp = t_1;
	elseif (z <= 3.3e+19)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(3.13060547623 * y + (-(-N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]))), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+14], t$95$1, If[LessEqual[z, 3.3e+19], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e14 or 3.3e19 < z

   1. Initial program 58.3%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, \color{blue}{y}, -1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, \mathsf{neg}\left(\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2.2e14 < z < 3.3e19

   1. Initial program 58.3%

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

Alternative 2: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(3.13060547623, y, -\left(-t \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{if}\;z \leq -2400000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(z \cdot z\right) \cdot z\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma 3.13060547623 y (- (- (* t (/ y (* z z)))))))))
   (if (<= z -2400000000000.0)
     t_1
     (if (<= z 3.3e+19)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+ (* (* (* z z) z) z) 0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(3.13060547623, y, -(-(t * (y / (z * z)))));
	double tmp;
	if (z <= -2400000000000.0) {
		tmp = t_1;
	} else if (z <= 3.3e+19) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / ((((z * z) * z) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(3.13060547623, y, Float64(-Float64(-Float64(t * Float64(y / Float64(z * z)))))))
	tmp = 0.0
	if (z <= -2400000000000.0)
		tmp = t_1;
	elseif (z <= 3.3e+19)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(z * z) * z) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(3.13060547623 * y + (-(-N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]))), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2400000000000.0], t$95$1, If[LessEqual[z, 3.3e+19], 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[(z * z), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e12 or 3.3e19 < z

   1. Initial program 58.3%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, \color{blue}{y}, -1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, \mathsf{neg}\left(\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2.4e12 < z < 3.3e19

   1. Initial program 58.3%

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

   2. Taylor expanded in z around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left({z}^{2} \cdot z\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(z \cdot z\right) \cdot z\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      5. lower-*.f64 57.2

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

   4. Applied rewrites 57.2%

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

Alternative 3: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(3.13060547623, y, -\left(-t \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{if}\;z \leq -2400000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 235000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\mathsf{fma}\left(\mathsf{fma}\left(31.4690115749, z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma 3.13060547623 y (- (- (* t (/ y (* z z)))))))))
   (if (<= z -2400000000.0)
     t_1
     (if (<= z 235000000000.0)
       (fma
        y
        (/
         (fma (fma t z a) z b)
         (fma (fma 31.4690115749 z 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 = x + fma(3.13060547623, y, -(-(t * (y / (z * z)))));
	double tmp;
	if (z <= -2400000000.0) {
		tmp = t_1;
	} else if (z <= 235000000000.0) {
		tmp = fma(y, (fma(fma(t, z, a), z, b) / fma(fma(31.4690115749, z, 11.9400905721), z, 0.607771387771)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(3.13060547623, y, Float64(-Float64(-Float64(t * Float64(y / Float64(z * z)))))))
	tmp = 0.0
	if (z <= -2400000000.0)
		tmp = t_1;
	elseif (z <= 235000000000.0)
		tmp = fma(y, Float64(fma(fma(t, z, a), z, b) / fma(fma(31.4690115749, z, 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[(x + N[(3.13060547623 * y + (-(-N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]))), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2400000000.0], t$95$1, If[LessEqual[z, 235000000000.0], N[(y * N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / N[(N[(31.4690115749 * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e9 or 2.35e11 < z

   1. Initial program 58.3%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, \color{blue}{y}, -1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, \mathsf{neg}\left(\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -2.4e9 < z < 2.35e11

   1. Initial program 58.3%

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

   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 64.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.1

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

   7. Applied rewrites 64.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 64.8%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 60.4

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

   12. Applied rewrites 60.4%

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

Alternative 4: 93.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(3.13060547623, y, -\left(-t \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7200000000:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(a, z, b\right)}{\left(11.9400905721 - -31.4690115749 \cdot z\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma 3.13060547623 y (- (- (* t (/ y (* z z)))))))))
   (if (<= z -3.8e+35)
     t_1
     (if (<= z 7200000000.0)
       (+
        x
        (/
         (* y (fma a z b))
         (+ (* (- 11.9400905721 (* -31.4690115749 z)) z) 0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(3.13060547623, y, -(-(t * (y / (z * z)))));
	double tmp;
	if (z <= -3.8e+35) {
		tmp = t_1;
	} else if (z <= 7200000000.0) {
		tmp = x + ((y * fma(a, z, b)) / (((11.9400905721 - (-31.4690115749 * z)) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(3.13060547623, y, Float64(-Float64(-Float64(t * Float64(y / Float64(z * z)))))))
	tmp = 0.0
	if (z <= -3.8e+35)
		tmp = t_1;
	elseif (z <= 7200000000.0)
		tmp = Float64(x + Float64(Float64(y * fma(a, z, b)) / Float64(Float64(Float64(11.9400905721 - Float64(-31.4690115749 * z)) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(3.13060547623 * y + (-(-N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]))), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e+35], t$95$1, If[LessEqual[z, 7200000000.0], N[(x + N[(N[(y * N[(a * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(11.9400905721 - N[(-31.4690115749 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8e35 or 7.2e9 < z

   1. Initial program 58.3%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, \color{blue}{y}, -1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, \mathsf{neg}\left(\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -3.8e35 < z < 7.2e9

   1. Initial program 58.3%

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

   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 64.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 64.2

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

   10. Applied rewrites 64.2%

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

Alternative 5: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(3.13060547623, y, -\left(-t \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{if}\;z \leq -0.051:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7200000000:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(a, z, b\right)}{11.9400905721 \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma 3.13060547623 y (- (- (* t (/ y (* z z)))))))))
   (if (<= z -0.051)
     t_1
     (if (<= z 7200000000.0)
       (+ x (/ (* y (fma a z b)) (+ (* 11.9400905721 z) 0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + fma(3.13060547623, y, -(-(t * (y / (z * z)))));
	double tmp;
	if (z <= -0.051) {
		tmp = t_1;
	} else if (z <= 7200000000.0) {
		tmp = x + ((y * fma(a, z, b)) / ((11.9400905721 * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(3.13060547623, y, Float64(-Float64(-Float64(t * Float64(y / Float64(z * z)))))))
	tmp = 0.0
	if (z <= -0.051)
		tmp = t_1;
	elseif (z <= 7200000000.0)
		tmp = Float64(x + Float64(Float64(y * fma(a, z, b)) / Float64(Float64(11.9400905721 * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(3.13060547623 * y + (-(-N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]))), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.051], t$95$1, If[LessEqual[z, 7200000000.0], N[(x + N[(N[(y * N[(a * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(11.9400905721 * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0509999999999999967 or 7.2e9 < z

   1. Initial program 58.3%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, \color{blue}{y}, -1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, \mathsf{neg}\left(\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -0.0509999999999999967 < z < 7.2e9

   1. Initial program 58.3%

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

   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 64.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 64.2

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

   10. Applied rewrites 64.2%

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

   11. Taylor expanded in z around 0

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

       1. lower-*.f64 63.2

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

   13. Applied rewrites 63.2%

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

Alternative 6: 86.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \mathsf{fma}\left(3.13060547623, y, -\left(-t \cdot \frac{y}{z \cdot z}\right)\right)\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 135000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(31.4690115749, z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (fma 3.13060547623 y (- (- (* t (/ y (* z z)))))))))
   (if (<= z -1.65e+25)
     t_1
     (if (<= z 135000.0)
       (fma
        y
        (/ b (fma (fma 31.4690115749 z 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 = x + fma(3.13060547623, y, -(-(t * (y / (z * z)))));
	double tmp;
	if (z <= -1.65e+25) {
		tmp = t_1;
	} else if (z <= 135000.0) {
		tmp = fma(y, (b / fma(fma(31.4690115749, z, 11.9400905721), z, 0.607771387771)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + fma(3.13060547623, y, Float64(-Float64(-Float64(t * Float64(y / Float64(z * z)))))))
	tmp = 0.0
	if (z <= -1.65e+25)
		tmp = t_1;
	elseif (z <= 135000.0)
		tmp = fma(y, Float64(b / fma(fma(31.4690115749, z, 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[(x + N[(3.13060547623 * y + (-(-N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]))), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.65e+25], t$95$1, If[LessEqual[z, 135000.0], N[(y * N[(b / N[(N[(31.4690115749 * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6500000000000001e25 or 135000 < z

   1. Initial program 58.3%

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

   2. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\frac{313060547623}{100000000000} \cdot y + \color{blue}{-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, \color{blue}{y}, -1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, \mathsf{neg}\left(\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{313060547623}{100000000000}, y, -\frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right) \]

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -1.6500000000000001e25 < z < 135000

   1. Initial program 58.3%

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

   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 64.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.1

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

   7. Applied rewrites 64.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 64.8%

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

Alternative 7: 84.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{b}{\mathsf{fma}\left(\mathsf{fma}\left(31.4690115749, z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.8e+35)
   (fma 3.13060547623 y x)
   (if (<= z 8.2e+17)
     (fma y (/ b (fma (fma 31.4690115749 z 11.9400905721) z 0.607771387771)) 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 <= -3.8e+35) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 8.2e+17) {
		tmp = fma(y, (b / fma(fma(31.4690115749, z, 11.9400905721), z, 0.607771387771)), x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.8e+35)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 8.2e+17)
		tmp = fma(y, Float64(b / fma(fma(31.4690115749, z, 11.9400905721), z, 0.607771387771)), x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.8e+35], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 8.2e+17], N[(y * N[(b / N[(N[(31.4690115749 * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8e35 or 8.2e17 < z

   1. Initial program 58.3%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

   if -3.8e35 < z < 8.2e17

   1. Initial program 58.3%

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

   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 64.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.1

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

   7. Applied rewrites 64.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 64.8%

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

Alternative 8: 83.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 1.6453555072203998 \cdot b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.8e+35)
   (fma 3.13060547623 y x)
   (if (<= z 8.2e+17)
     (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 <= -3.8e+35) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 8.2e+17) {
		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 <= -3.8e+35)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 8.2e+17)
		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, -3.8e+35], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 8.2e+17], 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 < -3.8e35 or 8.2e17 < z

   1. Initial program 58.3%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

   if -3.8e35 < z < 8.2e17

   1. Initial program 58.3%

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

   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 64.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval 64.1

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

   7. Applied rewrites 64.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 64.8%

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

   10. Taylor expanded in z around 0

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

       1. lower-*.f64 60.8

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

   12. Applied rewrites 60.8%

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

Alternative 9: 83.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(b, 1.6453555072203998 \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.8e+35)
   (fma 3.13060547623 y x)
   (if (<= z 8.2e+17)
     (fma b (* 1.6453555072203998 y) x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.8e+35) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 8.2e+17) {
		tmp = fma(b, (1.6453555072203998 * y), x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.8e+35)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 8.2e+17)
		tmp = fma(b, Float64(1.6453555072203998 * y), x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.8e+35], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 8.2e+17], N[(b * N[(1.6453555072203998 * y), $MachinePrecision] + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8e35 or 8.2e17 < z

   1. Initial program 58.3%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

   if -3.8e35 < z < 8.2e17

   1. Initial program 58.3%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 60.8

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

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 60.8

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

   6. Applied rewrites 60.8%

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

Alternative 10: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b \cdot y\right) \cdot 1.6453555072203998\\ t_2 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* b y) 1.6453555072203998))
        (t_2
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_2 -1e+37)
     t_1
     (if (<= t_2 5e+115)
       (fma 3.13060547623 y x)
       (if (<= t_2 INFINITY) t_1 (fma 3.13060547623 y x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (b * y) * 1.6453555072203998;
	double t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_2 <= -1e+37) {
		tmp = t_1;
	} else if (t_2 <= 5e+115) {
		tmp = fma(3.13060547623, y, x);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(b * y) * 1.6453555072203998)
	t_2 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -1e+37)
		tmp = t_1;
	elseif (t_2 <= 5e+115)
		tmp = fma(3.13060547623, y, x);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b * y), $MachinePrecision] * 1.6453555072203998), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+37], t$95$1, If[LessEqual[t$95$2, 5e+115], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(3.13060547623 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -9.99999999999999954e36 or 5.00000000000000008e115 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 58.3%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 60.8

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 22.0

         \[\leadsto \left(b \cdot y\right) \cdot 1.6453555072203998 \]

   7. Applied rewrites 22.0%

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

   if -9.99999999999999954e36 < (/.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))) < 5.00000000000000008e115 or +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 58.3%

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

   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 63.3

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

   4. Applied rewrites 63.3%

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

Alternative 11: 63.3% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.3%

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

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 63.3

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

4. Applied rewrites 63.3%

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

Alternative 12: 21.7% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 3.13060547623d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

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

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 63.3

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

4. Applied rewrites 63.3%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 21.7

      \[\leadsto 3.13060547623 \cdot y \]

7. Applied rewrites 21.7%

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

Reproduce

herbie shell --seed 2025142 
(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))))