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

Percentage Accurate: 69.0% → 99.6%

Time: 4.6s

Alternatives: 8

Speedup: 4.5×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      1e+305)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (- z -6.012459259764103) z 3.350343815022304))
    y
    x)
   (+ x (* 0.0692910599291889 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+305) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((z - -6.012459259764103), z, 3.350343815022304)), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+305)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(z - -6.012459259764103), z, 3.350343815022304)), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(z - -6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 9.9999999999999994e304

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   3. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, y, x\right)} \]

   if 9.9999999999999994e304 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 78.9

         \[\leadsto x + 0.0692910599291889 \cdot \color{blue}{y} \]

   4. Applied rewrites 78.9%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \mathsf{fma}\left(-0.14954831483277858, y, 0.14677053705526136 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + 0.4917317610505968 \cdot \frac{1}{z}\right) - 0.4166096748901212 \cdot \frac{1}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (+ x (* y (* (fma 0.0692910599291889 z 0.07512208616047561) (/ 1.0 z))))
   (if (<= z 3.6e-9)
     (+
      x
      (fma
       0.08333333333333323
       y
       (* z (fma -0.14954831483277858 y (* 0.14677053705526136 y)))))
     (+
      x
      (*
       y
       (-
        (+ 0.0692910599291889 (* 0.4917317610505968 (/ 1.0 z)))
        (* 0.4166096748901212 (/ 1.0 z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = x + (y * (fma(0.0692910599291889, z, 0.07512208616047561) * (1.0 / z)));
	} else if (z <= 3.6e-9) {
		tmp = x + fma(0.08333333333333323, y, (z * fma(-0.14954831483277858, y, (0.14677053705526136 * y))));
	} else {
		tmp = x + (y * ((0.0692910599291889 + (0.4917317610505968 * (1.0 / z))) - (0.4166096748901212 * (1.0 / z))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = Float64(x + Float64(y * Float64(fma(0.0692910599291889, z, 0.07512208616047561) * Float64(1.0 / z))));
	elseif (z <= 3.6e-9)
		tmp = Float64(x + fma(0.08333333333333323, y, Float64(z * fma(-0.14954831483277858, y, Float64(0.14677053705526136 * y)))));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(0.0692910599291889 + Float64(0.4917317610505968 * Float64(1.0 / z))) - Float64(0.4166096748901212 * Float64(1.0 / z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(x + N[(y * N[(N[(0.0692910599291889 * z + 0.07512208616047561), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e-9], N[(x + N[(0.08333333333333323 * y + N[(z * N[(-0.14954831483277858 * y + N[(0.14677053705526136 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(0.0692910599291889 + N[(0.4917317610505968 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.4166096748901212 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      6. lower-*.f64 53.2

         \[\leadsto x + \frac{\mathsf{fma}\left(0.0692910599291889, y \cdot z, 0.4917317610505968 \cdot y\right) - 0.4166096748901212 \cdot y}{z} \]

   7. Applied rewrites 53.2%

      \[\leadsto x + \frac{\mathsf{fma}\left(0.0692910599291889, y \cdot z, 0.4917317610505968 \cdot y\right) - 0.4166096748901212 \cdot y}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      2. sub-negate-rev N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)}{z} \]

      4. add-flip N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)\right)}{z} \]

      5. associate--r- N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      6. distribute-neg-in N/A

         \[\leadsto x + \frac{\left(\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      7. sub-negate-rev N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      8. sub-flip-reverse N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      9. remove-double-neg N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      11. *-commutative N/A

          \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(z \cdot y\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      12. associate-*r* N/A

          \[\leadsto x + \frac{\left(\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      14. lift-*.f64 N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      15. distribute-lft-neg-out N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot y\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      16. distribute-rgt-out N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      17. lower-fma.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y, z \cdot \frac{692910599291889}{10000000000000000} + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right), \frac{307332350656623}{625000000000000} \cdot y\right)}{z} \]

   9. Applied rewrites 53.3%

      \[\leadsto x + \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(0.0692910599291889, z, -0.4166096748901212\right), 0.4917317610505968 \cdot y\right)}{z} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right), \frac{307332350656623}{625000000000000} \cdot y\right)}{z} \]

       2. mult-flip N/A

          \[\leadsto x + \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right), \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{\color{blue}{z}} \]

       3. lift-fma.f64 N/A

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) + \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{z} \]

       4. lift-*.f64 N/A

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) + \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{z} \]

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

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot y\right) \cdot \frac{1}{z} \]

       6. *-commutative N/A

          \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot y - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot y\right) \cdot \frac{1}{z} \]

       7. distribute-rgt-out-- N/A

          \[\leadsto x + \left(y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right)\right) \cdot \frac{1}{z} \]

       8. associate-*l* N/A

          \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{\color{blue}{z}}\right) \]

       11. lift-fma.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{z}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{z}\right) \]

       13. associate--l+ N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{-4166096748901211929300981260567}{10000000000000000000000000000000} - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]

       14. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{-4166096748901211929300981260567}{10000000000000000000000000000000} - \frac{-307332350656623}{625000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       15. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       16. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       18. lower-fma.f64 N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       19. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       20. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{z}\right) \]

       21. lower-/.f64 N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{z}\right) \]

       22. metadata-eval 64.6

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \frac{1}{z}\right) \]

   11. Applied rewrites 64.6%

       \[\leadsto x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

   if -5.5 < z < 3.6e-9

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   3. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), \frac{z}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)} \cdot y, \mathsf{fma}\left(\frac{0.279195317918525}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, y, x\right)\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, \color{blue}{y}, z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \mathsf{fma}\left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416}, y, \frac{307332350656623}{2093964884388940} \cdot y\right)\right) \]

      5. lower-*.f64 66.2

         \[\leadsto x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \mathsf{fma}\left(-0.14954831483277858, y, 0.14677053705526136 \cdot y\right)\right) \]

   6. Applied rewrites 66.2%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \mathsf{fma}\left(-0.14954831483277858, y, 0.14677053705526136 \cdot y\right)\right)} \]

   if 3.6e-9 < z

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto x + y \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{1}{z}}\right) \]

      2. lower--.f64 N/A

         \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{1}{z}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{\color{blue}{1}}{z}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{1}{z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{1}{z}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{1}{\color{blue}{z}}\right) \]

      7. lower-/.f64 64.7

         \[\leadsto x + y \cdot \left(\left(0.0692910599291889 + 0.4917317610505968 \cdot \frac{1}{z}\right) - 0.4166096748901212 \cdot \frac{1}{z}\right) \]

   7. Applied rewrites 64.7%

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.0692910599291889 + 0.4917317610505968 \cdot \frac{1}{z}\right) - 0.4166096748901212 \cdot \frac{1}{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \mathsf{fma}\left(-0.14954831483277858, y, 0.14677053705526136 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(y, 0.0692910599291889, 0.07512208616047561 \cdot \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (+ x (* y (* (fma 0.0692910599291889 z 0.07512208616047561) (/ 1.0 z))))
   (if (<= z 3.6e-9)
     (+
      x
      (fma
       0.08333333333333323
       y
       (* z (fma -0.14954831483277858 y (* 0.14677053705526136 y)))))
     (+ x (fma y 0.0692910599291889 (* 0.07512208616047561 (/ y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = x + (y * (fma(0.0692910599291889, z, 0.07512208616047561) * (1.0 / z)));
	} else if (z <= 3.6e-9) {
		tmp = x + fma(0.08333333333333323, y, (z * fma(-0.14954831483277858, y, (0.14677053705526136 * y))));
	} else {
		tmp = x + fma(y, 0.0692910599291889, (0.07512208616047561 * (y / z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = Float64(x + Float64(y * Float64(fma(0.0692910599291889, z, 0.07512208616047561) * Float64(1.0 / z))));
	elseif (z <= 3.6e-9)
		tmp = Float64(x + fma(0.08333333333333323, y, Float64(z * fma(-0.14954831483277858, y, Float64(0.14677053705526136 * y)))));
	else
		tmp = Float64(x + fma(y, 0.0692910599291889, Float64(0.07512208616047561 * Float64(y / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(x + N[(y * N[(N[(0.0692910599291889 * z + 0.07512208616047561), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e-9], N[(x + N[(0.08333333333333323 * y + N[(z * N[(-0.14954831483277858 * y + N[(0.14677053705526136 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.0692910599291889 + N[(0.07512208616047561 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      6. lower-*.f64 53.2

         \[\leadsto x + \frac{\mathsf{fma}\left(0.0692910599291889, y \cdot z, 0.4917317610505968 \cdot y\right) - 0.4166096748901212 \cdot y}{z} \]

   7. Applied rewrites 53.2%

      \[\leadsto x + \frac{\mathsf{fma}\left(0.0692910599291889, y \cdot z, 0.4917317610505968 \cdot y\right) - 0.4166096748901212 \cdot y}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      2. sub-negate-rev N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)}{z} \]

      4. add-flip N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)\right)}{z} \]

      5. associate--r- N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      6. distribute-neg-in N/A

         \[\leadsto x + \frac{\left(\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      7. sub-negate-rev N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      8. sub-flip-reverse N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      9. remove-double-neg N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      11. *-commutative N/A

          \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(z \cdot y\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      12. associate-*r* N/A

          \[\leadsto x + \frac{\left(\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      14. lift-*.f64 N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      15. distribute-lft-neg-out N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot y\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      16. distribute-rgt-out N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      17. lower-fma.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y, z \cdot \frac{692910599291889}{10000000000000000} + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right), \frac{307332350656623}{625000000000000} \cdot y\right)}{z} \]

   9. Applied rewrites 53.3%

      \[\leadsto x + \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(0.0692910599291889, z, -0.4166096748901212\right), 0.4917317610505968 \cdot y\right)}{z} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right), \frac{307332350656623}{625000000000000} \cdot y\right)}{z} \]

       2. mult-flip N/A

          \[\leadsto x + \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right), \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{\color{blue}{z}} \]

       3. lift-fma.f64 N/A

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) + \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{z} \]

       4. lift-*.f64 N/A

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) + \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{z} \]

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

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot y\right) \cdot \frac{1}{z} \]

       6. *-commutative N/A

          \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot y - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot y\right) \cdot \frac{1}{z} \]

       7. distribute-rgt-out-- N/A

          \[\leadsto x + \left(y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right)\right) \cdot \frac{1}{z} \]

       8. associate-*l* N/A

          \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{\color{blue}{z}}\right) \]

       11. lift-fma.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{z}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{z}\right) \]

       13. associate--l+ N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{-4166096748901211929300981260567}{10000000000000000000000000000000} - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]

       14. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{-4166096748901211929300981260567}{10000000000000000000000000000000} - \frac{-307332350656623}{625000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       15. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       16. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       18. lower-fma.f64 N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       19. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       20. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{z}\right) \]

       21. lower-/.f64 N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{z}\right) \]

       22. metadata-eval 64.6

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \frac{1}{z}\right) \]

   11. Applied rewrites 64.6%

       \[\leadsto x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

   if -5.5 < z < 3.6e-9

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   3. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), \frac{z}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)} \cdot y, \mathsf{fma}\left(\frac{0.279195317918525}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, y, x\right)\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, \color{blue}{y}, z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y + \frac{307332350656623}{2093964884388940} \cdot y\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\frac{279195317918525}{3350343815022304}, y, z \cdot \mathsf{fma}\left(\frac{-1678650474502018223880473708075}{11224803678858206361900017468416}, y, \frac{307332350656623}{2093964884388940} \cdot y\right)\right) \]

      5. lower-*.f64 66.2

         \[\leadsto x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \mathsf{fma}\left(-0.14954831483277858, y, 0.14677053705526136 \cdot y\right)\right) \]

   6. Applied rewrites 66.2%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(0.08333333333333323, y, z \cdot \mathsf{fma}\left(-0.14954831483277858, y, 0.14677053705526136 \cdot y\right)\right)} \]

   if 3.6e-9 < z

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. associate--l+ N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{\left(\frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \left(\color{blue}{\frac{307332350656623}{625000000000000} \cdot \frac{y}{z}} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(y \cdot \frac{692910599291889}{10000000000000000} + \left(\color{blue}{\frac{307332350656623}{625000000000000} \cdot \frac{y}{z}} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{692910599291889}{10000000000000000}}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{y}{z} \cdot \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{y}{z}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{y}{z}\right) \]

      13. metadata-eval 64.7

          \[\leadsto x + \mathsf{fma}\left(y, 0.0692910599291889, 0.07512208616047561 \cdot \frac{y}{z}\right) \]

   6. Applied rewrites 64.7%

      \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{0.0692910599291889}, 0.07512208616047561 \cdot \frac{y}{z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \frac{1}{z}\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(y, 0.0692910599291889, 0.07512208616047561 \cdot \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (+ x (* y (* (fma 0.0692910599291889 z 0.07512208616047561) (/ 1.0 z))))
   (if (<= z 3.6e-9)
     (+ x (* 0.08333333333333323 y))
     (+ x (fma y 0.0692910599291889 (* 0.07512208616047561 (/ y z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = x + (y * (fma(0.0692910599291889, z, 0.07512208616047561) * (1.0 / z)));
	} else if (z <= 3.6e-9) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = x + fma(y, 0.0692910599291889, (0.07512208616047561 * (y / z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = Float64(x + Float64(y * Float64(fma(0.0692910599291889, z, 0.07512208616047561) * Float64(1.0 / z))));
	elseif (z <= 3.6e-9)
		tmp = Float64(x + Float64(0.08333333333333323 * y));
	else
		tmp = Float64(x + fma(y, 0.0692910599291889, Float64(0.07512208616047561 * Float64(y / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(x + N[(y * N[(N[(0.0692910599291889 * z + 0.07512208616047561), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e-9], N[(x + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * 0.0692910599291889 + N[(0.07512208616047561 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      6. lower-*.f64 53.2

         \[\leadsto x + \frac{\mathsf{fma}\left(0.0692910599291889, y \cdot z, 0.4917317610505968 \cdot y\right) - 0.4166096748901212 \cdot y}{z} \]

   7. Applied rewrites 53.2%

      \[\leadsto x + \frac{\mathsf{fma}\left(0.0692910599291889, y \cdot z, 0.4917317610505968 \cdot y\right) - 0.4166096748901212 \cdot y}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} \]

      2. sub-negate-rev N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y \cdot z, \frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)}{z} \]

      3. lift-fma.f64 N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)}{z} \]

      4. add-flip N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)\right)}{z} \]

      5. associate--r- N/A

         \[\leadsto x + \frac{\mathsf{neg}\left(\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      6. distribute-neg-in N/A

         \[\leadsto x + \frac{\left(\mathsf{neg}\left(\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y - \frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      7. sub-negate-rev N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      8. sub-flip-reverse N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000} \cdot y\right)\right)\right)\right)}{z} \]

      9. remove-double-neg N/A

         \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      11. *-commutative N/A

          \[\leadsto x + \frac{\left(\frac{692910599291889}{10000000000000000} \cdot \left(z \cdot y\right) + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      12. associate-*r* N/A

          \[\leadsto x + \frac{\left(\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      14. lift-*.f64 N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      15. distribute-lft-neg-out N/A

          \[\leadsto x + \frac{\left(\left(z \cdot \frac{692910599291889}{10000000000000000}\right) \cdot y + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot y\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      16. distribute-rgt-out N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right)\right) + \frac{307332350656623}{625000000000000} \cdot y}{z} \]

      17. lower-fma.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y, z \cdot \frac{692910599291889}{10000000000000000} + \left(\mathsf{neg}\left(\frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right), \frac{307332350656623}{625000000000000} \cdot y\right)}{z} \]

   9. Applied rewrites 53.3%

      \[\leadsto x + \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(0.0692910599291889, z, -0.4166096748901212\right), 0.4917317610505968 \cdot y\right)}{z} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right), \frac{307332350656623}{625000000000000} \cdot y\right)}{z} \]

       2. mult-flip N/A

          \[\leadsto x + \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right), \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{\color{blue}{z}} \]

       3. lift-fma.f64 N/A

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) + \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{z} \]

       4. lift-*.f64 N/A

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) + \frac{307332350656623}{625000000000000} \cdot y\right) \cdot \frac{1}{z} \]

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

          \[\leadsto x + \left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot y\right) \cdot \frac{1}{z} \]

       6. *-commutative N/A

          \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot y - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right) \cdot y\right) \cdot \frac{1}{z} \]

       7. distribute-rgt-out-- N/A

          \[\leadsto x + \left(y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right)\right) \cdot \frac{1}{z} \]

       8. associate-*l* N/A

          \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{\color{blue}{z}}\right) \]

       11. lift-fma.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{z}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right) - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right) \cdot \frac{1}{z}\right) \]

       13. associate--l+ N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{-4166096748901211929300981260567}{10000000000000000000000000000000} - \left(\mathsf{neg}\left(\frac{307332350656623}{625000000000000}\right)\right)\right)\right) \cdot \frac{1}{z}\right) \]

       14. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{-4166096748901211929300981260567}{10000000000000000000000000000000} - \frac{-307332350656623}{625000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       15. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       16. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto x + y \cdot \left(\left(\frac{692910599291889}{10000000000000000} \cdot z + \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \cdot \frac{1}{z}\right) \]

       18. lower-fma.f64 N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       19. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{1}{z}\right) \]

       20. metadata-eval N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{z}\right) \]

       21. lower-/.f64 N/A

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{z}\right) \]

       22. metadata-eval 64.6

           \[\leadsto x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \frac{1}{z}\right) \]

   11. Applied rewrites 64.6%

       \[\leadsto x + y \cdot \left(\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right) \cdot \color{blue}{\frac{1}{z}}\right) \]

   if -5.5 < z < 3.6e-9

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 80.2

         \[\leadsto x + 0.08333333333333323 \cdot \color{blue}{y} \]

   4. Applied rewrites 80.2%

      \[\leadsto x + \color{blue}{0.08333333333333323 \cdot y} \]

   if 3.6e-9 < z

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. associate--l+ N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{\left(\frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \left(\color{blue}{\frac{307332350656623}{625000000000000} \cdot \frac{y}{z}} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(y \cdot \frac{692910599291889}{10000000000000000} + \left(\color{blue}{\frac{307332350656623}{625000000000000} \cdot \frac{y}{z}} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{692910599291889}{10000000000000000}}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{y}{z} \cdot \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{y}{z}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{y}{z}\right) \]

      13. metadata-eval 64.7

          \[\leadsto x + \mathsf{fma}\left(y, 0.0692910599291889, 0.07512208616047561 \cdot \frac{y}{z}\right) \]

   6. Applied rewrites 64.7%

      \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{0.0692910599291889}, 0.07512208616047561 \cdot \frac{y}{z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \mathsf{fma}\left(y, 0.0692910599291889, 0.07512208616047561 \cdot \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (fma y 0.0692910599291889 (* 0.07512208616047561 (/ y z))))))
   (if (<= z -5.5)
     t_0
     (if (<= z 3.6e-9) (+ x (* 0.08333333333333323 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + fma(y, 0.0692910599291889, (0.07512208616047561 * (y / z)));
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 3.6e-9) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + fma(y, 0.0692910599291889, Float64(0.07512208616047561 * Float64(y / z))))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 3.6e-9)
		tmp = Float64(x + Float64(0.08333333333333323 * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * 0.0692910599291889 + N[(0.07512208616047561 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 3.6e-9], N[(x + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 3.6e-9 < z

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \color{blue}{\frac{y}{z}}\right) \]

      6. lower-/.f64 64.2

         \[\leadsto x + \left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{\color{blue}{z}}\right) \]

   4. Applied rewrites 64.2%

      \[\leadsto x + \color{blue}{\left(\mathsf{fma}\left(0.0692910599291889, y, 0.4917317610505968 \cdot \frac{y}{z}\right) - 0.4166096748901212 \cdot \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, y, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \color{blue}{\frac{4166096748901211929300981260567}{10000000000000000000000000000000}} \cdot \frac{y}{z}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x + \left(\left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      4. associate--l+ N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \color{blue}{\left(\frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \left(\color{blue}{\frac{307332350656623}{625000000000000} \cdot \frac{y}{z}} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(y \cdot \frac{692910599291889}{10000000000000000} + \left(\color{blue}{\frac{307332350656623}{625000000000000} \cdot \frac{y}{z}} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{692910599291889}{10000000000000000}}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right) \]

      10. distribute-rgt-out-- N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \frac{y}{z} \cdot \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{y}{z}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000}, \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right) \cdot \frac{y}{z}\right) \]

      13. metadata-eval 64.7

          \[\leadsto x + \mathsf{fma}\left(y, 0.0692910599291889, 0.07512208616047561 \cdot \frac{y}{z}\right) \]

   6. Applied rewrites 64.7%

      \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{0.0692910599291889}, 0.07512208616047561 \cdot \frac{y}{z}\right) \]

   if -5.5 < z < 3.6e-9

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 80.2

         \[\leadsto x + 0.08333333333333323 \cdot \color{blue}{y} \]

   4. Applied rewrites 80.2%

      \[\leadsto x + \color{blue}{0.08333333333333323 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + 0.0692910599291889 \cdot y\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* 0.0692910599291889 y))))
   (if (<= z -5.5)
     t_0
     (if (<= z 3.6e-9) (+ x (* 0.08333333333333323 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (0.0692910599291889 * y);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 3.6e-9) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (0.0692910599291889d0 * y)
    if (z <= (-5.5d0)) then
        tmp = t_0
    else if (z <= 3.6d-9) then
        tmp = x + (0.08333333333333323d0 * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (0.0692910599291889 * y);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 3.6e-9) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (0.0692910599291889 * y)
	tmp = 0
	if z <= -5.5:
		tmp = t_0
	elif z <= 3.6e-9:
		tmp = x + (0.08333333333333323 * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(0.0692910599291889 * y))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 3.6e-9)
		tmp = Float64(x + Float64(0.08333333333333323 * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (0.0692910599291889 * y);
	tmp = 0.0;
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 3.6e-9)
		tmp = x + (0.08333333333333323 * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 3.6e-9], N[(x + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 3.6e-9 < z

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 78.9

         \[\leadsto x + 0.0692910599291889 \cdot \color{blue}{y} \]

   4. Applied rewrites 78.9%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   if -5.5 < z < 3.6e-9

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 80.2

         \[\leadsto x + 0.08333333333333323 \cdot \color{blue}{y} \]

   4. Applied rewrites 80.2%

      \[\leadsto x + \color{blue}{0.08333333333333323 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889 \cdot z, z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      1e+305)
   (fma
    (fma (* 0.0692910599291889 z) z 0.279195317918525)
    (/ y (fma (- z -6.012459259764103) z 3.350343815022304))
    x)
   (+ x (* 0.0692910599291889 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+305) {
		tmp = fma(fma((0.0692910599291889 * z), z, 0.279195317918525), (y / fma((z - -6.012459259764103), z, 3.350343815022304)), x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+305)
		tmp = fma(fma(Float64(0.0692910599291889 * z), z, 0.279195317918525), Float64(y / fma(Float64(z - -6.012459259764103), z, 3.350343815022304)), x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(N[(0.0692910599291889 * z), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * N[(y / N[(N[(z - -6.012459259764103), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 9.9999999999999994e304

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   3. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), \frac{z}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)} \cdot y, \mathsf{fma}\left(\frac{0.279195317918525}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, y, x\right)\right)} \]

   5. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z}, z, \frac{11167812716741}{40000000000000}\right), \frac{y}{\mathsf{fma}\left(z - \frac{-6012459259764103}{1000000000000000}, z, \frac{104698244219447}{31250000000000}\right)}, x\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 72.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889 \cdot \color{blue}{z}, z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, x\right) \]

   8. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0692910599291889 \cdot z}, z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z - -6.012459259764103, z, 3.350343815022304\right)}, x\right) \]

   if 9.9999999999999994e304 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 69.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 78.9

         \[\leadsto x + 0.0692910599291889 \cdot \color{blue}{y} \]

   4. Applied rewrites 78.9%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x + 0.0692910599291889 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* 0.0692910599291889 y)))

C

double code(double x, double y, double z) {
	return x + (0.0692910599291889 * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (0.0692910599291889d0 * y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (0.0692910599291889 * y);
}

Python

def code(x, y, z):
	return x + (0.0692910599291889 * y)

Julia

function code(x, y, z)
	return Float64(x + Float64(0.0692910599291889 * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (0.0692910599291889 * y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

2. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
3. Step-by-step derivation

   1. lower-*.f64 78.9

      \[\leadsto x + 0.0692910599291889 \cdot \color{blue}{y} \]

4. Applied rewrites 78.9%

   \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64
  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))