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

Percentage Accurate: 69.0% → 98.7%

Time: 11.3s

Alternatives: 16

Speedup: 2.5×

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: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 6.012459259764103\right) \cdot z + 3.350343815022304\\ t_1 := x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{t\_0}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, 0.4917317610505968, \mathsf{fma}\left(z \cdot z, 0.0692910599291889, 0.279195317918525\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304))
        (t_1
         (+
          x
          (/
           (*
            y
            (+
             (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
             0.279195317918525))
           t_0))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
      (/ y (fma (+ 6.012459259764103 z) z 3.350343815022304)))
     (if (<= t_1 1e+295)
       (+
        x
        (/
         (*
          y
          (fma
           z
           0.4917317610505968
           (fma (* z z) 0.0692910599291889 0.279195317918525)))
         t_0))
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = ((z + 6.012459259764103) * z) + 3.350343815022304;
	double t_1 = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / t_0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * (y / fma((6.012459259764103 + z), z, 3.350343815022304));
	} else if (t_1 <= 1e+295) {
		tmp = x + ((y * fma(z, 0.4917317610505968, fma((z * z), 0.0692910599291889, 0.279195317918525))) / t_0);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)
	t_1 = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / t_0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) * Float64(y / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)));
	elseif (t_1 <= 1e+295)
		tmp = Float64(x + Float64(Float64(y * fma(z, 0.4917317610505968, fma(Float64(z * z), 0.0692910599291889, 0.279195317918525))) / t_0));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * N[(y / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+295], N[(x + N[(N[(y * N[(z * 0.4917317610505968 + N[(N[(z * z), $MachinePrecision] * 0.0692910599291889 + 0.279195317918525), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)))) < -inf.0

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right) \cdot y}}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right) \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right) \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right) + \frac{11167812716741}{40000000000000}\right)} \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z, z, \frac{11167812716741}{40000000000000}\right)} \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot \color{blue}{\frac{y}{\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)}} \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\color{blue}{z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right) + \frac{104698244219447}{31250000000000}}} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\color{blue}{\left(\frac{6012459259764103}{1000000000000000} + z\right) \cdot z} + \frac{104698244219447}{31250000000000}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}} \]

      13. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -inf.0 < (+.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)))) < 9.9999999999999998e294

   1. Initial program 99.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(\color{blue}{z \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\left(\frac{307332350656623}{625000000000000} + z \cdot \frac{692910599291889}{10000000000000000}\right)} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. distribute-lft-in N/A

         \[\leadsto x + \frac{y \cdot \left(\color{blue}{\left(z \cdot \frac{307332350656623}{625000000000000} + z \cdot \left(z \cdot \frac{692910599291889}{10000000000000000}\right)\right)} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. associate-+l+ N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \frac{307332350656623}{625000000000000} + \left(z \cdot \left(z \cdot \frac{692910599291889}{10000000000000000}\right) + \frac{11167812716741}{40000000000000}\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, \frac{307332350656623}{625000000000000}, z \cdot \left(z \cdot \frac{692910599291889}{10000000000000000}\right) + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(z, \frac{307332350656623}{625000000000000}, z \cdot \color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000}\right)} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. associate-*r* N/A

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(z, \frac{307332350656623}{625000000000000}, \color{blue}{\left(z \cdot z\right) \cdot \frac{692910599291889}{10000000000000000}} + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. lower-fma.f64 N/A

          \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(z, \frac{307332350656623}{625000000000000}, \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{692910599291889}{10000000000000000}, \frac{11167812716741}{40000000000000}\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 9.9999999999999998e294 < (+.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 4.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

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

Alternative 2: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+128} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+156} \lor \neg \left(t\_0 \leq 10^{+295}\right)\right):\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* 0.0692910599291889 y)
     (if (or (<= t_0 -1e+128) (not (or (<= t_0 2e+156) (not (<= t_0 1e+295)))))
       (* 0.08333333333333323 y)
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0692910599291889 * y;
	} else if ((t_0 <= -1e+128) || !((t_0 <= 2e+156) || !(t_0 <= 1e+295))) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0692910599291889 * y);
	elseif ((t_0 <= -1e+128) || !((t_0 <= 2e+156) || !(t_0 <= 1e+295)))
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y), $MachinePrecision], If[Or[LessEqual[t$95$0, -1e+128], N[Not[Or[LessEqual[t$95$0, 2e+156], N[Not[LessEqual[t$95$0, 1e+295]], $MachinePrecision]]], $MachinePrecision]], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.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))) < -inf.0

   1. Initial program 7.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.3%

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

   if -inf.0 < (/.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))) < -1.0000000000000001e128 or 2e156 < (/.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))) < 9.9999999999999998e294

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. lower-fma.f64 91.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.7%

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

   if -1.0000000000000001e128 < (/.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))) < 2e156 or 9.9999999999999998e294 < (/.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 65.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 87.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -1 \cdot 10^{+128} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 2 \cdot 10^{+156} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+295}\right)\right):\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 6.012459259764103\right) \cdot z + 3.350343815022304\\ t_1 := x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{t\_0}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;x + \frac{t\_2 \cdot y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304))
        (t_1
         (+
          x
          (/
           (*
            y
            (+
             (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
             0.279195317918525))
           t_0)))
        (t_2
         (fma
          (fma 0.0692910599291889 z 0.4917317610505968)
          z
          0.279195317918525)))
   (if (<= t_1 -4e-60)
     (fma t_2 (/ y (fma (+ 6.012459259764103 z) z 3.350343815022304)) x)
     (if (<= t_1 1e+295)
       (+ x (/ (* t_2 y) t_0))
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = ((z + 6.012459259764103) * z) + 3.350343815022304;
	double t_1 = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / t_0);
	double t_2 = fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525);
	double tmp;
	if (t_1 <= -4e-60) {
		tmp = fma(t_2, (y / fma((6.012459259764103 + z), z, 3.350343815022304)), x);
	} else if (t_1 <= 1e+295) {
		tmp = x + ((t_2 * y) / t_0);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)
	t_1 = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / t_0))
	t_2 = fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525)
	tmp = 0.0
	if (t_1 <= -4e-60)
		tmp = fma(t_2, Float64(y / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), x);
	elseif (t_1 <= 1e+295)
		tmp = Float64(x + Float64(Float64(t_2 * y) / t_0));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-60], N[(t$95$2 * N[(y / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+295], N[(x + N[(N[(t$95$2 * y), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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)))) < -3.9999999999999999e-60

   1. Initial program 90.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}, \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, x\right)} \]

   4. Applied rewrites 99.6%

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

   if -3.9999999999999999e-60 < (+.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)))) < 9.9999999999999998e294

   1. Initial program 99.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. lower-*.f64 99.6

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

      4. lift-+.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. lift-*.f64 N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. lower-fma.f64 99.6

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

      7. lift-+.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lift-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. *-commutative N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 9.9999999999999998e294 < (+.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 4.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)\\ t_2 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{y}{t\_2}, x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+295}:\\ \;\;\;\;x + \frac{t\_1 \cdot y}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          x
          (/
           (*
            y
            (+
             (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
             0.279195317918525))
           (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
        (t_1
         (fma
          (fma 0.0692910599291889 z 0.4917317610505968)
          z
          0.279195317918525))
        (t_2 (fma (+ 6.012459259764103 z) z 3.350343815022304)))
   (if (<= t_0 -4e-60)
     (fma t_1 (/ y t_2) x)
     (if (<= t_0 1e+295)
       (+ x (/ (* t_1 y) t_2))
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
	double t_1 = fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525);
	double t_2 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double tmp;
	if (t_0 <= -4e-60) {
		tmp = fma(t_1, (y / t_2), x);
	} else if (t_0 <= 1e+295) {
		tmp = x + ((t_1 * y) / t_2);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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)))
	t_1 = fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525)
	t_2 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	tmp = 0.0
	if (t_0 <= -4e-60)
		tmp = fma(t_1, Float64(y / t_2), x);
	elseif (t_0 <= 1e+295)
		tmp = Float64(x + Float64(Float64(t_1 * y) / t_2));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision]}, Block[{t$95$2 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-60], N[(t$95$1 * N[(y / t$95$2), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 1e+295], N[(x + N[(N[(t$95$1 * y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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)))) < -3.9999999999999999e-60

   1. Initial program 90.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}, \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, x\right)} \]

   4. Applied rewrites 99.6%

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

   if -3.9999999999999999e-60 < (+.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)))) < 9.9999999999999998e294

   1. Initial program 99.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. lower-*.f64 99.6

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

      4. lift-+.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. lift-*.f64 N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. lower-fma.f64 99.6

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

      7. lift-+.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lift-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. *-commutative N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. lower-fma.f64 99.6

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

      11. lift-+.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}} \]

      13. lower-fma.f64 99.6

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

      14. lift-+.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)} \]

      15. +-commutative N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\mathsf{fma}\left(\color{blue}{\frac{6012459259764103}{1000000000000000} + z}, z, \frac{104698244219447}{31250000000000}\right)} \]

      16. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 9.9999999999999998e294 < (+.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 4.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

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

Alternative 5: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot 0.0692910599291889\right) \cdot y + 0.4917317610505968 \cdot y, \frac{z}{t\_0}, \mathsf{fma}\left(y, \frac{0.279195317918525}{t\_0}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ 6.012459259764103 z) z 3.350343815022304)))
   (if (<=
        (+
         x
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
        1e+295)
     (fma
      (+ (* (* z 0.0692910599291889) y) (* 0.4917317610505968 y))
      (/ z t_0)
      (fma y (/ 0.279195317918525 t_0) x))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double t_0 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+295) {
		tmp = fma((((z * 0.0692910599291889) * y) + (0.4917317610505968 * y)), (z / t_0), fma(y, (0.279195317918525 / t_0), x));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	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+295)
		tmp = fma(Float64(Float64(Float64(z * 0.0692910599291889) * y) + Float64(0.4917317610505968 * y)), Float64(z / t_0), fma(y, Float64(0.279195317918525 / t_0), x));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, 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+295], N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] + N[(0.4917317610505968 * y), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$0), $MachinePrecision] + N[(y * N[(0.279195317918525 / t$95$0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $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)))) < 9.9999999999999998e294

   1. Initial program 96.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. distribute-lft-in N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right) + y \cdot \frac{11167812716741}{40000000000000}}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)} + y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. associate-*r* N/A

         \[\leadsto x + \frac{\color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z} + y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z + \color{blue}{\frac{11167812716741}{40000000000000} \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lift-+.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. *-commutative N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-fma.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      13. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525 \cdot y\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z + \frac{11167812716741}{40000000000000} \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)} + x \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + \left(\frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, \frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot y, \frac{z}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, \mathsf{fma}\left(y, \frac{0.279195317918525}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot y}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{307332350656623}{625000000000000}\right)}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y + \frac{307332350656623}{625000000000000} \cdot y}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y + \frac{307332350656623}{625000000000000} \cdot y}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y} + \frac{307332350656623}{625000000000000} \cdot y, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000}\right)} \cdot y + \frac{307332350656623}{625000000000000} \cdot y, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000}\right)} \cdot y + \frac{307332350656623}{625000000000000} \cdot y, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      9. lower-*.f64 98.3

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

   8. Applied rewrites 98.3%

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

   if 9.9999999999999998e294 < (+.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 4.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot 0.0692910599291889, y, 0.4917317610505968 \cdot y\right), \frac{z}{t\_0}, \mathsf{fma}\left(y, \frac{0.279195317918525}{t\_0}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ 6.012459259764103 z) z 3.350343815022304)))
   (if (<=
        (+
         x
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
        1e+295)
     (fma
      (fma (* z 0.0692910599291889) y (* 0.4917317610505968 y))
      (/ z t_0)
      (fma y (/ 0.279195317918525 t_0) x))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double t_0 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+295) {
		tmp = fma(fma((z * 0.0692910599291889), y, (0.4917317610505968 * y)), (z / t_0), fma(y, (0.279195317918525 / t_0), x));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	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+295)
		tmp = fma(fma(Float64(z * 0.0692910599291889), y, Float64(0.4917317610505968 * y)), Float64(z / t_0), fma(y, Float64(0.279195317918525 / t_0), x));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, 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+295], N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] * y + N[(0.4917317610505968 * y), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$0), $MachinePrecision] + N[(y * N[(0.279195317918525 / t$95$0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $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)))) < 9.9999999999999998e294

   1. Initial program 96.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. distribute-lft-in N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right) + y \cdot \frac{11167812716741}{40000000000000}}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)} + y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. associate-*r* N/A

         \[\leadsto x + \frac{\color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z} + y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z + \color{blue}{\frac{11167812716741}{40000000000000} \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lift-+.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. *-commutative N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-fma.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      13. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525 \cdot y\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z + \frac{11167812716741}{40000000000000} \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)} + x \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + \left(\frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, \frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot y, \frac{z}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, \mathsf{fma}\left(y, \frac{0.279195317918525}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot y}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot z + \frac{307332350656623}{625000000000000}\right)}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot z\right) \cdot y + \frac{307332350656623}{625000000000000} \cdot y}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000} \cdot z, y, \frac{307332350656623}{625000000000000} \cdot y\right)}, \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}}, y, \frac{307332350656623}{625000000000000} \cdot y\right), \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}}, y, \frac{307332350656623}{625000000000000} \cdot y\right), \frac{z}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, \mathsf{fma}\left(y, \frac{\frac{11167812716741}{40000000000000}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}, x\right)\right) \]

      8. lower-*.f64 98.2

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

   8. Applied rewrites 98.2%

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

   if 9.9999999999999998e294 < (+.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 4.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)\\ \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot y, \frac{z}{t\_0}, \mathsf{fma}\left(y, \frac{0.279195317918525}{t\_0}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ 6.012459259764103 z) z 3.350343815022304)))
   (if (<=
        (+
         x
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
        1e+295)
     (fma
      (* (fma 0.0692910599291889 z 0.4917317610505968) y)
      (/ z t_0)
      (fma y (/ 0.279195317918525 t_0) x))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double t_0 = fma((6.012459259764103 + z), z, 3.350343815022304);
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 1e+295) {
		tmp = fma((fma(0.0692910599291889, z, 0.4917317610505968) * y), (z / t_0), fma(y, (0.279195317918525 / t_0), x));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(6.012459259764103 + z), z, 3.350343815022304)
	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+295)
		tmp = fma(Float64(fma(0.0692910599291889, z, 0.4917317610505968) * y), Float64(z / t_0), fma(y, Float64(0.279195317918525 / t_0), x));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]}, 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+295], N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * y), $MachinePrecision] * N[(z / t$95$0), $MachinePrecision] + N[(y * N[(0.279195317918525 / t$95$0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $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)))) < 9.9999999999999998e294

   1. Initial program 96.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. lift-+.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. distribute-lft-in N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right) + y \cdot \frac{11167812716741}{40000000000000}}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. lift-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z\right)} + y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. associate-*r* N/A

         \[\leadsto x + \frac{\color{blue}{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z} + y \cdot \frac{11167812716741}{40000000000000}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)\right) \cdot z + \color{blue}{\frac{11167812716741}{40000000000000} \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lower-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lift-+.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. lift-*.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. *-commutative N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-fma.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(y \cdot \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)}, z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      13. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525 \cdot y\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000} \cdot y\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z + \frac{11167812716741}{40000000000000} \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)} + x \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + \left(\frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right) \cdot \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + \left(\frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), \frac{z}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, \frac{\frac{11167812716741}{40000000000000} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x\right)} \]

   6. Applied rewrites 98.2%

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

   if 9.9999999999999998e294 < (+.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 4.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{0.0692910599291889}{x}, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8e+18)
   (fma 0.0692910599291889 y x)
   (if (<= z 3.8e+109)
     (fma
      (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
      (/ y (fma (+ 6.012459259764103 z) z 3.350343815022304))
      x)
     (* (fma y (/ 0.0692910599291889 x) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8e+18) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 3.8e+109) {
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), (y / fma((6.012459259764103 + z), z, 3.350343815022304)), x);
	} else {
		tmp = fma(y, (0.0692910599291889 / x), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8e+18)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 3.8e+109)
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), Float64(y / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), x);
	else
		tmp = Float64(fma(y, Float64(0.0692910599291889 / x), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8e+18], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 3.8e+109], N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * N[(y / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y * N[(0.0692910599291889 / x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8e18

   1. Initial program 39.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -8e18 < z < 3.80000000000000039e109

   1. Initial program 97.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}, \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, x\right)} \]

   4. Applied rewrites 99.5%

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

   if 3.80000000000000039e109 < z

   1. Initial program 14.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{692910599291889}{10000000000000000} \cdot \frac{y}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x}, 0.0692910599291889, 1\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(y, \frac{0.0692910599291889}{x}, 1\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, y, \left(y \cdot 0.0007936505811533442\right) \cdot z\right), z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (if (<= z 5.8e-17)
     (fma
      (fma -0.00277777777751721 y (* (* y 0.0007936505811533442) z))
      z
      (fma 0.08333333333333323 y x))
     (fma 0.07512208616047561 (/ y z) (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else if (z <= 5.8e-17) {
		tmp = fma(fma(-0.00277777777751721, y, ((y * 0.0007936505811533442) * z)), z, fma(0.08333333333333323, y, x));
	} else {
		tmp = fma(0.07512208616047561, (y / z), fma(0.0692910599291889, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	elseif (z <= 5.8e-17)
		tmp = fma(fma(-0.00277777777751721, y, Float64(Float64(y * 0.0007936505811533442) * z)), z, fma(0.08333333333333323, y, x));
	else
		tmp = fma(0.07512208616047561, Float64(y / z), fma(0.0692910599291889, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.8e-17], N[(N[(-0.00277777777751721 * y + N[(N[(y * 0.0007936505811533442), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z + N[(0.08333333333333323 * y + x), $MachinePrecision]), $MachinePrecision], N[(0.07512208616047561 * N[(y / z), $MachinePrecision] + N[(0.0692910599291889 * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 43.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 97.4%

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

   if -5.4000000000000004 < z < 5.8000000000000006e-17

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) \cdot z} + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y, z, x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, y, \left(y \cdot 0.0007936505811533442\right) \cdot z\right), z, \mathsf{fma}\left(0.08333333333333323, y, x\right)\right)} \]

   if 5.8000000000000006e-17 < z

   1. Initial program 47.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 98.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

   8. Applied rewrites 99.2%

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

Alternative 10: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.07512208616047561, \frac{y}{z}, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (if (<= z 5.8e-17)
     (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)
     (fma 0.07512208616047561 (/ y z) (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else if (z <= 5.8e-17) {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	} else {
		tmp = fma(0.07512208616047561, (y / z), fma(0.0692910599291889, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	elseif (z <= 5.8e-17)
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	else
		tmp = fma(0.07512208616047561, Float64(y / z), fma(0.0692910599291889, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.8e-17], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(0.07512208616047561 * N[(y / z), $MachinePrecision] + N[(0.0692910599291889 * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 43.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 97.4%

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

   if -5.4000000000000004 < z < 5.8000000000000006e-17

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.3

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{-0.00277777777751721}, z, 0.08333333333333323\right), x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)} \]

   if 5.8000000000000006e-17 < z

   1. Initial program 47.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 98.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

   8. Applied rewrites 99.2%

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

Alternative 11: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.4) (not (<= z 5.8e-17)))
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.4) || !(z <= 5.8e-17)) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.4) || !(z <= 5.8e-17))
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.4], N[Not[LessEqual[z, 5.8e-17]], $MachinePrecision]], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5.8000000000000006e-17 < z

   1. Initial program 44.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.1%

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

   if -5.4000000000000004 < z < 5.8000000000000006e-17

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.3

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{-0.00277777777751721}, z, 0.08333333333333323\right), x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.4) (not (<= z 5.8e-17)))
   (fma 0.0692910599291889 y x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.4) || !(z <= 5.8e-17)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.4) || !(z <= 5.8e-17))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.4], N[Not[LessEqual[z, 5.8e-17]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5.8000000000000006e-17 < z

   1. Initial program 44.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 97.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -5.4000000000000004 < z < 5.8000000000000006e-17

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.3

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{-0.00277777777751721}, z, 0.08333333333333323\right), x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-32}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-234}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-74}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e-32)
   (* 1.0 x)
   (if (<= x -8e-234)
     (* 0.08333333333333323 y)
     (if (<= x 8e-74) (* 0.0692910599291889 y) (* 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-32) {
		tmp = 1.0 * x;
	} else if (x <= -8e-234) {
		tmp = 0.08333333333333323 * y;
	} else if (x <= 8e-74) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.9d-32)) then
        tmp = 1.0d0 * x
    else if (x <= (-8d-234)) then
        tmp = 0.08333333333333323d0 * y
    else if (x <= 8d-74) then
        tmp = 0.0692910599291889d0 * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-32) {
		tmp = 1.0 * x;
	} else if (x <= -8e-234) {
		tmp = 0.08333333333333323 * y;
	} else if (x <= 8e-74) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.9e-32:
		tmp = 1.0 * x
	elif x <= -8e-234:
		tmp = 0.08333333333333323 * y
	elif x <= 8e-74:
		tmp = 0.0692910599291889 * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e-32)
		tmp = Float64(1.0 * x);
	elseif (x <= -8e-234)
		tmp = Float64(0.08333333333333323 * y);
	elseif (x <= 8e-74)
		tmp = Float64(0.0692910599291889 * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.9e-32)
		tmp = 1.0 * x;
	elseif (x <= -8e-234)
		tmp = 0.08333333333333323 * y;
	elseif (x <= 8e-74)
		tmp = 0.0692910599291889 * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e-32], N[(1.0 * x), $MachinePrecision], If[LessEqual[x, -8e-234], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[x, 8e-74], N[(0.0692910599291889 * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9000000000000001e-32 or 7.99999999999999966e-74 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 88.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{692910599291889}{10000000000000000} \cdot \frac{y}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto 1 \cdot x \]

   if -3.9000000000000001e-32 < x < -7.9999999999999997e-234

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. lower-fma.f64 79.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.4%

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

   if -7.9999999999999997e-234 < x < 7.99999999999999966e-74

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 78.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

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

Alternative 14: 98.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.4) (not (<= z 5.8e-17)))
   (fma 0.0692910599291889 y x)
   (fma 0.08333333333333323 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.4) || !(z <= 5.8e-17)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(0.08333333333333323, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.4) || !(z <= 5.8e-17))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(0.08333333333333323, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.4], N[Not[LessEqual[z, 5.8e-17]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5.8000000000000006e-17 < z

   1. Initial program 44.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 97.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -5.4000000000000004 < z < 5.8000000000000006e-17

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. lower-fma.f64 98.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333323, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \lor \neg \left(z \leq 5.8 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-32} \lor \neg \left(x \leq 8 \cdot 10^{-74}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.9e-32) (not (<= x 8e-74)))
   (* 1.0 x)
   (* 0.0692910599291889 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e-32) || !(x <= 8e-74)) {
		tmp = 1.0 * x;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	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) :: tmp
    if ((x <= (-3.9d-32)) .or. (.not. (x <= 8d-74))) then
        tmp = 1.0d0 * x
    else
        tmp = 0.0692910599291889d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e-32) || !(x <= 8e-74)) {
		tmp = 1.0 * x;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.9e-32) or not (x <= 8e-74):
		tmp = 1.0 * x
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.9e-32) || !(x <= 8e-74))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.9e-32) || ~((x <= 8e-74)))
		tmp = 1.0 * x;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.9e-32], N[Not[LessEqual[x, 8e-74]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000001e-32 or 7.99999999999999966e-74 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 88.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{692910599291889}{10000000000000000} \cdot \frac{y}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 70.9%

       \[\leadsto 1 \cdot x \]

   if -3.9000000000000001e-32 < x < 7.99999999999999966e-74

   1. Initial program 71.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 67.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.9%

      \[\leadsto 0.0692910599291889 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-32} \lor \neg \left(x \leq 8 \cdot 10^{-74}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.2% accurate, 7.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = 0.0692910599291889d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

   2. lower-fma.f64 79.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

5. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 34.7%

   \[\leadsto 0.0692910599291889 \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y)
          (- (/ (* 0.40462203869992125 y) (* z z)) x))))
   (if (< z -8120153.652456675)
     t_0
     (if (< z 6.576118972787377e+20)
       (+
        x
        (*
         (*
          y
          (+
           (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
           0.279195317918525))
         (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	double tmp;
	if (z < -8120153.652456675) {
		tmp = t_0;
	} else if (z < 6.576118972787377e+20) {
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	} 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 = (((0.07512208616047561d0 / z) + 0.0692910599291889d0) * y) - (((0.40462203869992125d0 * y) / (z * z)) - x)
    if (z < (-8120153.652456675d0)) then
        tmp = t_0
    else if (z < 6.576118972787377d+20) then
        tmp = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) * (1.0d0 / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	double tmp;
	if (z < -8120153.652456675) {
		tmp = t_0;
	} else if (z < 6.576118972787377e+20) {
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x)
	tmp = 0
	if z < -8120153.652456675:
		tmp = t_0
	elif z < 6.576118972787377e+20:
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889) * y) - Float64(Float64(Float64(0.40462203869992125 * y) / Float64(z * z)) - x))
	tmp = 0.0
	if (z < -8120153.652456675)
		tmp = t_0;
	elseif (z < 6.576118972787377e+20)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * Float64(1.0 / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	tmp = 0.0;
	if (z < -8120153.652456675)
		tmp = t_0;
	elseif (z < 6.576118972787377e+20)
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(0.40462203869992125 * y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -8120153.652456675], t$95$0, If[Less[z, 6.576118972787377e+20], N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -324806146098267/40000000) (- (* (+ (/ 7512208616047561/100000000000000000 z) 692910599291889/10000000000000000) y) (- (/ (* 323697630959937/800000000000000 y) (* z z)) x)) (if (< z 657611897278737700000) (+ x (* (* y (+ (* (+ (* z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000)) (/ 1 (+ (* (+ z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)))) (- (* (+ (/ 7512208616047561/100000000000000000 z) 692910599291889/10000000000000000) y) (- (/ (* 323697630959937/800000000000000 y) (* z z)) x)))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))