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

Percentage Accurate: 59.1% → 98.1%

Time: 5.2s

Alternatives: 12

Speedup: 3.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+40}:\\ \;\;\;\;\left(\mathsf{fma}\left(3.13060547623, y, x\right) + \mathsf{fma}\left(\frac{y}{z}, 11.1667541262, t \cdot t\_1\right)\right) - \mathsf{fma}\left(\frac{y \cdot -36.52704169880642}{z \cdot z}, 15.234687407, \mathsf{fma}\left(t\_1, 98.5170599679272, 47.69379582500642 \cdot \frac{y}{z}\right)\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(-\frac{36.52704169880642 + \frac{\left(-t\right) - 457.9610022158428}{z}}{z}\right) + 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ y (* z z))))
   (if (<= z -1.95e+40)
     (-
      (+ (fma 3.13060547623 y x) (fma (/ y z) 11.1667541262 (* t t_1)))
      (fma
       (/ (* y -36.52704169880642) (* z z))
       15.234687407
       (fma t_1 98.5170599679272 (* 47.69379582500642 (/ y z)))))
     (if (<= z 8.8e+30)
       (+
        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)))
       (fma
        y
        (+
         (- (/ (+ 36.52704169880642 (/ (- (- t) 457.9610022158428) z)) z))
         3.13060547623)
        x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y / (z * z);
	double tmp;
	if (z <= -1.95e+40) {
		tmp = (fma(3.13060547623, y, x) + fma((y / z), 11.1667541262, (t * t_1))) - fma(((y * -36.52704169880642) / (z * z)), 15.234687407, fma(t_1, 98.5170599679272, (47.69379582500642 * (y / z))));
	} else if (z <= 8.8e+30) {
		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 = fma(y, (-((36.52704169880642 + ((-t - 457.9610022158428) / z)) / z) + 3.13060547623), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y / Float64(z * z))
	tmp = 0.0
	if (z <= -1.95e+40)
		tmp = Float64(Float64(fma(3.13060547623, y, x) + fma(Float64(y / z), 11.1667541262, Float64(t * t_1))) - fma(Float64(Float64(y * -36.52704169880642) / Float64(z * z)), 15.234687407, fma(t_1, 98.5170599679272, Float64(47.69379582500642 * Float64(y / z)))));
	elseif (z <= 8.8e+30)
		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 = fma(y, Float64(Float64(-Float64(Float64(36.52704169880642 + Float64(Float64(Float64(-t) - 457.9610022158428) / z)) / z)) + 3.13060547623), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.95e+40], N[(N[(N[(3.13060547623 * y + x), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * 11.1667541262 + N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y * -36.52704169880642), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] * 15.234687407 + N[(t$95$1 * 98.5170599679272 + N[(47.69379582500642 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.8e+30], 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], N[(y * N[((-N[(N[(36.52704169880642 + N[(N[((-t) - 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.95e40

   1. Initial program 8.3%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 97.7%

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

   if -1.95e40 < z < 8.7999999999999999e30

   1. Initial program 98.4%

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

   if 8.7999999999999999e30 < z

   1. Initial program 9.3%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 19.2%

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

   5. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 96.9%

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

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{36.52704169880642 + \frac{\left(-t\right) - 457.9610022158428}{z}}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          y
          (+
           (- (/ (+ 36.52704169880642 (/ (- (- t) 457.9610022158428) z)) z))
           3.13060547623)
          x)))
   (if (<= z -1.95e+40)
     t_1
     (if (<= z 8.8e+30)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         (+
          (*
           (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
           z)
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (-((36.52704169880642 + ((-t - 457.9610022158428) / z)) / z) + 3.13060547623), x);
	double tmp;
	if (z <= -1.95e+40) {
		tmp = t_1;
	} else if (z <= 8.8e+30) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(Float64(36.52704169880642 + Float64(Float64(Float64(-t) - 457.9610022158428) / z)) / z)) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -1.95e+40)
		tmp = t_1;
	elseif (z <= 8.8e+30)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-N[(N[(36.52704169880642 + N[(N[((-t) - 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.95e+40], t$95$1, If[LessEqual[z, 8.8e+30], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e40 or 8.7999999999999999e30 < z

   1. Initial program 8.8%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 97.6%

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

   if -1.95e40 < z < 8.7999999999999999e30

   1. Initial program 98.4%

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

Alternative 3: 96.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{36.52704169880642 + \frac{\left(-t\right) - 457.9610022158428}{z}}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -1600000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          y
          (+
           (- (/ (+ 36.52704169880642 (/ (- (- t) 457.9610022158428) z)) z))
           3.13060547623)
          x)))
   (if (<= z -1600000.0)
     t_1
     (if (<= z 8.4e+29)
       (+ x (/ (* y (fma (fma t z a) z b)) 0.607771387771))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (-((36.52704169880642 + ((-t - 457.9610022158428) / z)) / z) + 3.13060547623), x);
	double tmp;
	if (z <= -1600000.0) {
		tmp = t_1;
	} else if (z <= 8.4e+29) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / 0.607771387771);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(Float64(36.52704169880642 + Float64(Float64(Float64(-t) - 457.9610022158428) / z)) / z)) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -1600000.0)
		tmp = t_1;
	elseif (z <= 8.4e+29)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / 0.607771387771));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-N[(N[(36.52704169880642 + N[(N[((-t) - 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1600000.0], t$95$1, If[LessEqual[z, 8.4e+29], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6e6 or 8.4000000000000006e29 < z

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 21.9%

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

   5. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 96.1%

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

   if -1.6e6 < z < 8.4000000000000006e29

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 95.9

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

   7. Applied rewrites 95.9%

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

Alternative 4: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \left(-\frac{36.52704169880642 + \frac{\left(-t\right) - 457.9610022158428}{z}}{z}\right) + 3.13060547623, x\right)\\ \mathbf{if}\;z \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          y
          (+
           (- (/ (+ 36.52704169880642 (/ (- (- t) 457.9610022158428) z)) z))
           3.13060547623)
          x)))
   (if (<= z -200000.0)
     t_1
     (if (<= z 8.4e+29)
       (+
        x
        (/
         (*
          y
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))
         0.607771387771))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (-((36.52704169880642 + ((-t - 457.9610022158428) / z)) / z) + 3.13060547623), x);
	double tmp;
	if (z <= -200000.0) {
		tmp = t_1;
	} else if (z <= 8.4e+29) {
		tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / 0.607771387771);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(-Float64(Float64(36.52704169880642 + Float64(Float64(Float64(-t) - 457.9610022158428) / z)) / z)) + 3.13060547623), x)
	tmp = 0.0
	if (z <= -200000.0)
		tmp = t_1;
	elseif (z <= 8.4e+29)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / 0.607771387771));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[((-N[(N[(36.52704169880642 + N[(N[((-t) - 457.9610022158428), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -200000.0], t$95$1, If[LessEqual[z, 8.4e+29], N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e5 or 8.4000000000000006e29 < z

   1. Initial program 13.3%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 96.0%

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

   if -2e5 < z < 8.4000000000000006e29

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 96.4%

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

Alternative 5: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -90000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -90000000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 4.6e+30)
     (+ x (/ (* y (fma (fma t z a) z b)) 0.607771387771))
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -90000000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 4.6e+30) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / 0.607771387771);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -90000000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 4.6e+30)
		tmp = Float64(x + Float64(Float64(y * fma(fma(t, z, a), z, b)) / 0.607771387771));
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -90000000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 4.6e+30], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e10 or 4.6e30 < z

   1. Initial program 12.2%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if -9e10 < z < 4.6e30

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 95.6

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

   7. Applied rewrites 95.6%

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

Alternative 6: 90.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -52000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-111}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(a, z, b\right)}{0.607771387771}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+18}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(t \cdot z\right) \cdot z + b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -52000000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 2.6e-111)
     (+ x (/ (* y (fma a z b)) 0.607771387771))
     (if (<= z 1.7e+18)
       (+ x (/ (* y (+ (* (* t z) z) b)) 0.607771387771))
       (fma 3.13060547623 y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -52000000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 2.6e-111) {
		tmp = x + ((y * fma(a, z, b)) / 0.607771387771);
	} else if (z <= 1.7e+18) {
		tmp = x + ((y * (((t * z) * z) + b)) / 0.607771387771);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -52000000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 2.6e-111)
		tmp = Float64(x + Float64(Float64(y * fma(a, z, b)) / 0.607771387771));
	elseif (z <= 1.7e+18)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(t * z) * z) + b)) / 0.607771387771));
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -52000000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 2.6e-111], N[(x + N[(N[(y * N[(a * z + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e+18], N[(x + N[(N[(y * N[(N[(N[(t * z), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e10 or 1.7e18 < z

   1. Initial program 13.5%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if -5.2e10 < z < 2.59999999999999982e-111

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if 2.59999999999999982e-111 < z < 1.7e18

   1. Initial program 99.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in t around inf

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

Alternative 7: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -52000000000:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+31}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(a, z, b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -52000000000.0)
   (fma 3.13060547623 y x)
   (if (<= z 6.5e+31)
     (+ x (/ (* y (fma a z b)) 0.607771387771))
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -52000000000.0) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 6.5e+31) {
		tmp = x + ((y * fma(a, z, b)) / 0.607771387771);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -52000000000.0)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 6.5e+31)
		tmp = Float64(x + Float64(Float64(y * fma(a, z, b)) / 0.607771387771));
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -52000000000.0], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 6.5e+31], N[(x + N[(N[(y * N[(a * z + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e10 or 6.5000000000000004e31 < z

   1. Initial program 12.0%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.2

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

   4. Applied rewrites 91.2%

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

   if -5.2e10 < z < 6.5000000000000004e31

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 89.9

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

   7. Applied rewrites 89.9%

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

Alternative 8: 83.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(1.6453555072203998 \cdot b, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.1e+19)
   (fma 3.13060547623 y x)
   (if (<= z 5e+16)
     (fma (* 1.6453555072203998 b) y x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.1e+19) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 5e+16) {
		tmp = fma((1.6453555072203998 * b), y, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.1e+19)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 5e+16)
		tmp = fma(Float64(1.6453555072203998 * b), y, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.1e+19], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 5e+16], N[(N[(1.6453555072203998 * b), $MachinePrecision] * y + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1e19 or 5e16 < z

   1. Initial program 12.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 90.8

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

   4. Applied rewrites 90.8%

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

   if -3.1e19 < z < 5e16

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 77.6

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

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)} \]
   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 77.6

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

   6. Applied rewrites 77.6%

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

Alternative 9: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+223}:\\ \;\;\;\;\left(1.6453555072203998 \cdot b\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 10^{+28}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(b \cdot y\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_1 -1e+223)
     (* (* 1.6453555072203998 b) y)
     (if (<= t_1 1e+28)
       x
       (if (<= t_1 INFINITY)
         (* (* b y) 1.6453555072203998)
         (fma 3.13060547623 y x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_1 <= -1e+223) {
		tmp = (1.6453555072203998 * b) * y;
	} else if (t_1 <= 1e+28) {
		tmp = x;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (b * y) * 1.6453555072203998;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -1e+223)
		tmp = Float64(Float64(1.6453555072203998 * b) * y);
	elseif (t_1 <= 1e+28)
		tmp = x;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(b * y) * 1.6453555072203998);
	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[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+223], N[(N[(1.6453555072203998 * b), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 1e+28], x, If[LessEqual[t$95$1, Infinity], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1.00000000000000005e223

   1. Initial program 79.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   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 54.3

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

   7. Applied rewrites 54.3%

      \[\leadsto \left(b \cdot y\right) \cdot \color{blue}{1.6453555072203998} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 54.3

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

   9. Applied rewrites 54.3%

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

   if -1.00000000000000005e223 < (/.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))) < 9.99999999999999958e27

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 60.9%

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

   if 9.99999999999999958e27 < (/.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 89.6%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 63.0

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

   4. Applied rewrites 63.0%

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

   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 47.5

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

   7. Applied rewrites 47.5%

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

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

   1. Initial program 0.0%

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

   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 97.4

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

   4. Applied rewrites 97.4%

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

Alternative 10: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1.6453555072203998 \cdot b\right) \cdot y\\ t_2 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+223}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+28}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 1.6453555072203998 b) y))
        (t_2
         (/
          (*
           y
           (+
            (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
            b))
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771))))
   (if (<= t_2 -1e+223)
     t_1
     (if (<= t_2 1e+28)
       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 = (1.6453555072203998 * b) * y;
	double t_2 = (y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771);
	double tmp;
	if (t_2 <= -1e+223) {
		tmp = t_1;
	} else if (t_2 <= 1e+28) {
		tmp = 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(1.6453555072203998 * b) * y)
	t_2 = Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -1e+223)
		tmp = t_1;
	elseif (t_2 <= 1e+28)
		tmp = 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[(1.6453555072203998 * b), $MachinePrecision] * y), $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+223], t$95$1, If[LessEqual[t$95$2, 1e+28], x, If[LessEqual[t$95$2, Infinity], t$95$1, N[(3.13060547623 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1.00000000000000005e223 or 9.99999999999999958e27 < (/.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 86.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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 62.1

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

   4. Applied rewrites 62.1%

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

   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 49.9

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

   7. Applied rewrites 49.9%

      \[\leadsto \left(b \cdot y\right) \cdot \color{blue}{1.6453555072203998} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 49.9

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

   9. Applied rewrites 49.9%

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

   if -1.00000000000000005e223 < (/.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))) < 9.99999999999999958e27

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 60.9%

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

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

   1. Initial program 86.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 19.6

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

   4. Applied rewrites 19.6%

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

Alternative 11: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
         b))
       (+
        (*
         (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
         z)
        0.607771387771))
      2e+52)
   x
   (fma 3.13060547623 y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)) <= 2e+52) {
		tmp = x;
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)) <= 2e+52)
		tmp = x;
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 2e+52], x, 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))) < 2e52

   1. Initial program 95.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 49.6%

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

   if 2e52 < (/.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 28.4%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 72.8

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

   4. Applied rewrites 72.8%

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

Alternative 12: 44.0% accurate, 52.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 44.0%

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

Reproduce

herbie shell --seed 2025119 
(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))))