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

Percentage Accurate: 58.8% → 97.9%

Time: 6.1s

Alternatives: 13

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: 58.8% 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: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot y, \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.1e+46)
   (fma
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    y
    x)
   (if (<= z 1.4e+36)
     (fma
      (* (fma (fma (fma (fma 3.13060547623 z 11.1667541262) z t) z a) z b) y)
      (/
       1.0
       (fma
        (fma (fma (- z -15.234687407) z 31.4690115749) z 11.9400905721)
        z
        0.607771387771))
      x)
     (fma
      (+
       3.13060547623
       (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.1e+46) {
		tmp = fma((-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), y, x);
	} else if (z <= 1.4e+36) {
		tmp = fma((fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) * y), (1.0 / fma(fma(fma((z - -15.234687407), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)), x);
	} else {
		tmp = fma((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.1e+46)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), y, x);
	elseif (z <= 1.4e+36)
		tmp = fma(Float64(fma(fma(fma(fma(3.13060547623, z, 11.1667541262), z, t), z, a), z, b) * y), Float64(1.0 / fma(fma(fma(Float64(z - -15.234687407), z, 31.4690115749), z, 11.9400905721), z, 0.607771387771)), x);
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.1e+46], N[(N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 1.4e+36], N[(N[(N[(N[(N[(N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] * y), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[(z - -15.234687407), $MachinePrecision] * z + 31.4690115749), $MachinePrecision] * z + 11.9400905721), $MachinePrecision] * z + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e46

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around -inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-+.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.1e46 < z < 1.4e36

   1. Initial program 58.8%

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

   3. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), z, a\right), z, b\right) \cdot y, \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}, x\right)} \]

   if 1.4e36 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. div-add-rev N/A

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

       5. lower-/.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. mult-flip-rev N/A

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

       10. lower-/.f64 56.2

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

   11. Applied rewrites 56.2%

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

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+35}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.5e+23)
   (fma
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    y
    x)
   (if (<= z 8e+35)
     (+
      x
      (/
       (* y (fma (fma t z a) z b))
       (+
        (*
         (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
         z)
        0.607771387771)))
     (fma
      (+
       3.13060547623
       (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.5e+23) {
		tmp = fma((-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), y, x);
	} else if (z <= 8e+35) {
		tmp = x + ((y * fma(fma(t, z, a), z, b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
	} else {
		tmp = fma((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.5e+23)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), y, x);
	elseif (z <= 8e+35)
		tmp = Float64(x + Float64(Float64(y * fma(fma(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(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.5e+23], N[(N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 8e+35], N[(x + N[(N[(y * N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5e23

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around -inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-+.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.5e23 < z < 7.9999999999999997e35

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 62.6

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

   4. Applied rewrites 62.6%

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

   if 7.9999999999999997e35 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. div-add-rev N/A

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

       5. lower-/.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. mult-flip-rev N/A

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

       10. lower-/.f64 56.2

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

   11. Applied rewrites 56.2%

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

Alternative 3: 96.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1300000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(11.1667541262, z, t\right), z, a\right), z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.32e+14)
   (fma
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    y
    x)
   (if (<= z 1300000.0)
     (fma y (/ (fma (fma (fma 11.1667541262 z t) z a) z b) 0.607771387771) x)
     (fma
      (+
       3.13060547623
       (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.32e+14) {
		tmp = fma((-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), y, x);
	} else if (z <= 1300000.0) {
		tmp = fma(y, (fma(fma(fma(11.1667541262, z, t), z, a), z, b) / 0.607771387771), x);
	} else {
		tmp = fma((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.32e+14)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), y, x);
	elseif (z <= 1300000.0)
		tmp = fma(y, Float64(fma(fma(fma(11.1667541262, z, t), z, a), z, b) / 0.607771387771), x);
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.32e+14], N[(N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 1300000.0], N[(y * N[(N[(N[(N[(11.1667541262 * z + t), $MachinePrecision] * z + a), $MachinePrecision] * z + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e14

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around -inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-+.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.32e14 < z < 1.3e6

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 60.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 60.0%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 56.0

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

   12. Applied rewrites 56.0%

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

   if 1.3e6 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. div-add-rev N/A

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

       5. lower-/.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. mult-flip-rev N/A

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

       10. lower-/.f64 56.2

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

   11. Applied rewrites 56.2%

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

Alternative 4: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1300000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \left(\frac{457.9610022158428 + t}{z \cdot z} - \frac{36.52704169880642}{z}\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.32e+14)
   (fma
    (+
     (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
     3.13060547623)
    y
    x)
   (if (<= z 1300000.0)
     (fma y (/ (fma (fma t z a) z b) 0.607771387771) x)
     (fma
      (+
       3.13060547623
       (- (/ (+ 457.9610022158428 t) (* z z)) (/ 36.52704169880642 z)))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.32e+14) {
		tmp = fma((-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), y, x);
	} else if (z <= 1300000.0) {
		tmp = fma(y, (fma(fma(t, z, a), z, b) / 0.607771387771), x);
	} else {
		tmp = fma((3.13060547623 + (((457.9610022158428 + t) / (z * z)) - (36.52704169880642 / z))), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.32e+14)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), y, x);
	elseif (z <= 1300000.0)
		tmp = fma(y, Float64(fma(fma(t, z, a), z, b) / 0.607771387771), x);
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 + t) / Float64(z * z)) - Float64(36.52704169880642 / z))), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.32e+14], N[(N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 1300000.0], N[(y * N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(457.9610022158428 + t), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.32e14

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around -inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-+.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.32e14 < z < 1.3e6

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 60.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 60.0%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 58.0

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

   12. Applied rewrites 58.0%

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

   if 1.3e6 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. div-add-rev N/A

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

       5. lower-/.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. mult-flip-rev N/A

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

       10. lower-/.f64 56.2

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

   11. Applied rewrites 56.2%

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

Alternative 5: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\left(-\frac{\left(-\frac{457.9610022158428 + t}{z}\right) + 36.52704169880642}{z}\right) + 3.13060547623, y, x\right)\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1300000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          (+
           (- (/ (+ (- (/ (+ 457.9610022158428 t) z)) 36.52704169880642) z))
           3.13060547623)
          y
          x)))
   (if (<= z -1.32e+14)
     t_1
     (if (<= z 1300000.0)
       (fma y (/ (fma (fma t z a) z b) 0.607771387771) x)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((-((-((457.9610022158428 + t) / z) + 36.52704169880642) / z) + 3.13060547623), y, x);
	double tmp;
	if (z <= -1.32e+14) {
		tmp = t_1;
	} else if (z <= 1300000.0) {
		tmp = fma(y, (fma(fma(t, z, a), z, b) / 0.607771387771), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(457.9610022158428 + t) / z)) + 36.52704169880642) / z)) + 3.13060547623), y, x)
	tmp = 0.0
	if (z <= -1.32e+14)
		tmp = t_1;
	elseif (z <= 1300000.0)
		tmp = fma(y, Float64(fma(fma(t, z, a), z, b) / 0.607771387771), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[((-N[(N[((-N[(N[(457.9610022158428 + t), $MachinePrecision] / z), $MachinePrecision]) + 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]) + 3.13060547623), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -1.32e+14], t$95$1, If[LessEqual[z, 1300000.0], N[(y * N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e14 or 1.3e6 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/l* N/A

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

   6. Applied rewrites 67.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in z around -inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-+.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.32e14 < z < 1.3e6

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 60.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 60.0%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 6: 92.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 2060000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t, z, a\right), z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.6e+31)
   (fma 3.13060547623 y x)
   (if (<= z 2060000000.0)
     (fma y (/ (fma (fma t z a) z b) 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 <= -3.6e+31) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 2060000000.0) {
		tmp = fma(y, (fma(fma(t, z, a), z, b) / 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 <= -3.6e+31)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 2060000000.0)
		tmp = fma(y, Float64(fma(fma(t, z, a), z, b) / 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, -3.6e+31], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 2060000000.0], N[(y * N[(N[(N[(t * z + a), $MachinePrecision] * z + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999996e31 or 2.06e9 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 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 62.2

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

   4. Applied rewrites 62.2%

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

   if -3.59999999999999996e31 < z < 2.06e9

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 60.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 60.0%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 58.0

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

   12. Applied rewrites 58.0%

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

Alternative 7: 90.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+35}:\\ \;\;\;\;x + \left(y \cdot \mathsf{fma}\left(a, z, b\right)\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -8e+21)
   (fma 3.13060547623 y x)
   (if (<= z 7.2e+35)
     (+ x (* (* y (fma a z b)) 1.6453555072203998))
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -8e+21) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 7.2e+35) {
		tmp = x + ((y * fma(a, z, b)) * 1.6453555072203998);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -8e+21)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 7.2e+35)
		tmp = Float64(x + Float64(Float64(y * fma(a, z, b)) * 1.6453555072203998));
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -8e+21], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 7.2e+35], N[(x + N[(N[(y * N[(a * z + b), $MachinePrecision]), $MachinePrecision] * 1.6453555072203998), $MachinePrecision]), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e21 or 7.2000000000000001e35 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 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 62.2

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

   4. Applied rewrites 62.2%

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

   if -8e21 < z < 7.2000000000000001e35

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. mult-flip N/A

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

   6. Applied rewrites 66.0%

      \[\leadsto x + \color{blue}{\left(y \cdot \mathsf{fma}\left(a, z, b\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z - -15.234687407, z, 31.4690115749\right), z, 11.9400905721\right), z, 0.607771387771\right)}} \]

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 60.1%

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

Alternative 8: 90.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(a, z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -8e+21)
   (fma 3.13060547623 y x)
   (if (<= z 7.2e+35)
     (fma y (/ (fma a z b) 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 <= -8e+21) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 7.2e+35) {
		tmp = fma(y, (fma(a, z, b) / 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 <= -8e+21)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 7.2e+35)
		tmp = fma(y, Float64(fma(a, z, b) / 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, -8e+21], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 7.2e+35], N[(y * N[(N[(a * z + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e21 or 7.2000000000000001e35 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 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 62.2

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

   4. Applied rewrites 62.2%

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

   if -8e21 < z < 7.2000000000000001e35

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 60.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 60.0%

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

Alternative 9: 82.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot y, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.6e+31)
   (fma 3.13060547623 y x)
   (if (<= z 1.55e-34)
     (fma (* b y) 1.6453555072203998 x)
     (fma 3.13060547623 y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.6e+31) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 1.55e-34) {
		tmp = fma((b * y), 1.6453555072203998, x);
	} else {
		tmp = fma(3.13060547623, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.6e+31)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 1.55e-34)
		tmp = fma(Float64(b * y), 1.6453555072203998, x);
	else
		tmp = fma(3.13060547623, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.6e+31], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 1.55e-34], N[(N[(b * y), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(3.13060547623 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999996e31 or 1.5499999999999999e-34 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 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 62.2

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

   4. Applied rewrites 62.2%

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

   if -3.59999999999999996e31 < z < 1.5499999999999999e-34

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 0

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

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

Alternative 10: 82.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(y, 1.6453555072203998 \cdot b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.6e+31)
   (fma 3.13060547623 y x)
   (if (<= z 1.55e-34)
     (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 <= -3.6e+31) {
		tmp = fma(3.13060547623, y, x);
	} else if (z <= 1.55e-34) {
		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 <= -3.6e+31)
		tmp = fma(3.13060547623, y, x);
	elseif (z <= 1.55e-34)
		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, -3.6e+31], N[(3.13060547623 * y + x), $MachinePrecision], If[LessEqual[z, 1.55e-34], 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 < -3.59999999999999996e31 or 1.5499999999999999e-34 < z

   1. Initial program 58.8%

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

   2. Taylor expanded in z around 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 62.2

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

   4. Applied rewrites 62.2%

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

   if -3.59999999999999996e31 < z < 1.5499999999999999e-34

   1. Initial program 58.8%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

Alternative 11: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(1.6453555072203998 \cdot y\right)\\ t_2 := \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (* 1.6453555072203998 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 -2000.0)
     t_1
     (if (<= t_2 4e+164)
       (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 * (1.6453555072203998 * 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 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 4e+164) {
		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(b * Float64(1.6453555072203998 * 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 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 4e+164)
		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[(b * N[(1.6453555072203998 * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 4e+164], 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))) < -2e3 or 4e164 < (/.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 58.8%

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

   2. Taylor expanded in b around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

   if -2e3 < (/.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))) < 4e164 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 58.8%

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

   2. Taylor expanded in z around 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 62.2

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

   4. Applied rewrites 62.2%

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

Alternative 12: 62.2% 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 58.8%

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

2. Taylor expanded in z around 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 62.2

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

4. Applied rewrites 62.2%

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

Alternative 13: 21.8% 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 58.8%

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

2. Taylor expanded in z around 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 62.2

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

4. Applied rewrites 62.2%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 21.8

      \[\leadsto 3.13060547623 \cdot y \]

7. Applied rewrites 21.8%

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

Reproduce

herbie shell --seed 2025134 
(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))))