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

Percentage Accurate: 69.6% → 98.9%

Time: 12.7s

Alternatives: 12

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

real(8) function code(x, y, z)
    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: 69.6% 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

real(8) function code(x, y, z)
    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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      2e+286)
   (/
    1.0
    (/
     1.0
     (fma
      y
      (/
       (fma z (fma z 0.0692910599291889 0.4917317610505968) 0.279195317918525)
       (fma z (+ z 6.012459259764103) 3.350343815022304))
      x)))
   (fma y 0.0692910599291889 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= 2e+286) {
		tmp = 1.0 / (1.0 / fma(y, (fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, (z + 6.012459259764103), 3.350343815022304)), x));
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= 2e+286)
		tmp = Float64(1.0 / Float64(1.0 / fma(y, Float64(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, Float64(z + 6.012459259764103), 3.350343815022304)), x)));
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], 2e+286], N[(1.0 / N[(1.0 / N[(y * N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < 2.00000000000000007e286

   1. Initial program 98.0%

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

      1. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)}}} \]

   if 2.00000000000000007e286 < (/.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 0.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -500000000:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;t\_0 \leq 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+286}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
            0.279195317918525))
          (+ (* z (+ z 6.012459259764103)) 3.350343815022304))))
   (if (<= t_0 -500000000.0)
     (* y 0.08333333333333323)
     (if (<= t_0 1e+54)
       (fma y 0.0692910599291889 x)
       (if (<= t_0 2e+286)
         (* y 0.08333333333333323)
         (fma y 0.0692910599291889 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = y * 0.08333333333333323;
	} else if (t_0 <= 1e+54) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (t_0 <= 2e+286) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= -500000000.0)
		tmp = Float64(y * 0.08333333333333323);
	elseif (t_0 <= 1e+54)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (t_0 <= 2e+286)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500000000.0], N[(y * 0.08333333333333323), $MachinePrecision], If[LessEqual[t$95$0, 1e+54], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[t$95$0, 2e+286], N[(y * 0.08333333333333323), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]]]

Derivation

1. Split input into 2 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))) < -5e8 or 1.0000000000000001e54 < (/.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))) < 2.00000000000000007e286

   1. Initial program 95.3%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 69.6

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

   8. Simplified 69.6%

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

   if -5e8 < (/.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))) < 1.0000000000000001e54 or 2.00000000000000007e286 < (/.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 59.2%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 92.0

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

   5. Simplified 92.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -500000000:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -500000000:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;t\_0 \leq 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+286}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
            0.279195317918525))
          (+ (* z (+ z 6.012459259764103)) 3.350343815022304))))
   (if (<= t_0 -500000000.0)
     (* y 0.08333333333333323)
     (if (<= t_0 1e+54)
       x
       (if (<= t_0 2e+286)
         (* y 0.08333333333333323)
         (* y 0.0692910599291889))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = y * 0.08333333333333323;
	} else if (t_0 <= 1e+54) {
		tmp = x;
	} else if (t_0 <= 2e+286) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = y * 0.0692910599291889;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * ((z * ((z * 0.0692910599291889d0) + 0.4917317610505968d0)) + 0.279195317918525d0)) / ((z * (z + 6.012459259764103d0)) + 3.350343815022304d0)
    if (t_0 <= (-500000000.0d0)) then
        tmp = y * 0.08333333333333323d0
    else if (t_0 <= 1d+54) then
        tmp = x
    else if (t_0 <= 2d+286) then
        tmp = y * 0.08333333333333323d0
    else
        tmp = y * 0.0692910599291889d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = y * 0.08333333333333323;
	} else if (t_0 <= 1e+54) {
		tmp = x;
	} else if (t_0 <= 2e+286) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = y * 0.0692910599291889;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)
	tmp = 0
	if t_0 <= -500000000.0:
		tmp = y * 0.08333333333333323
	elif t_0 <= 1e+54:
		tmp = x
	elif t_0 <= 2e+286:
		tmp = y * 0.08333333333333323
	else:
		tmp = y * 0.0692910599291889
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= -500000000.0)
		tmp = Float64(y * 0.08333333333333323);
	elseif (t_0 <= 1e+54)
		tmp = x;
	elseif (t_0 <= 2e+286)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = Float64(y * 0.0692910599291889);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	tmp = 0.0;
	if (t_0 <= -500000000.0)
		tmp = y * 0.08333333333333323;
	elseif (t_0 <= 1e+54)
		tmp = x;
	elseif (t_0 <= 2e+286)
		tmp = y * 0.08333333333333323;
	else
		tmp = y * 0.0692910599291889;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500000000.0], N[(y * 0.08333333333333323), $MachinePrecision], If[LessEqual[t$95$0, 1e+54], x, If[LessEqual[t$95$0, 2e+286], N[(y * 0.08333333333333323), $MachinePrecision], N[(y * 0.0692910599291889), $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))) < -5e8 or 1.0000000000000001e54 < (/.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))) < 2.00000000000000007e286

   1. Initial program 95.3%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 69.6

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

   8. Simplified 69.6%

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

   if -5e8 < (/.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))) < 1.0000000000000001e54

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.5%

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

   if 2.00000000000000007e286 < (/.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 0.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -500000000:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      2e+286)
   (fma
    (fma z (fma z 0.0692910599291889 0.4917317610505968) 0.279195317918525)
    (/ y (fma z (+ z 6.012459259764103) 3.350343815022304))
    x)
   (fma y 0.0692910599291889 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= 2e+286) {
		tmp = fma(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525), (y / fma(z, (z + 6.012459259764103), 3.350343815022304)), x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= 2e+286)
		tmp = fma(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525), Float64(y / fma(z, Float64(z + 6.012459259764103), 3.350343815022304)), x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], 2e+286], N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] * N[(y / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < 2.00000000000000007e286

   1. Initial program 98.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-lowering-+.f64 98.9

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

   4. Applied egg-rr 98.9%

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

   if 2.00000000000000007e286 < (/.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 0.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 + \frac{0.07512208616047561 + \frac{-0.4046220386999212}{z}}{z}, x\right)\\ \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.0005951669793454025, 0.0007936505811533442\right), -0.00277777777751721\right), 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          y
          (+
           0.0692910599291889
           (/ (+ 0.07512208616047561 (/ -0.4046220386999212 z)) z))
          x)))
   (if (<= z -5.4)
     t_0
     (if (<= z 3.5)
       (fma
        y
        (fma
         z
         (fma
          z
          (fma z -0.0005951669793454025 0.0007936505811533442)
          -0.00277777777751721)
         0.08333333333333323)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 + ((0.07512208616047561 + (-0.4046220386999212 / z)) / z)), x);
	double tmp;
	if (z <= -5.4) {
		tmp = t_0;
	} else if (z <= 3.5) {
		tmp = fma(y, fma(z, fma(z, fma(z, -0.0005951669793454025, 0.0007936505811533442), -0.00277777777751721), 0.08333333333333323), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 + Float64(Float64(0.07512208616047561 + Float64(-0.4046220386999212 / z)) / z)), x)
	tmp = 0.0
	if (z <= -5.4)
		tmp = t_0;
	elseif (z <= 3.5)
		tmp = fma(y, fma(z, fma(z, fma(z, -0.0005951669793454025, 0.0007936505811533442), -0.00277777777751721), 0.08333333333333323), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 + N[(N[(0.07512208616047561 + N[(-0.4046220386999212 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.4], t$95$0, If[LessEqual[z, 3.5], N[(y * N[(z * N[(z * N[(z * -0.0005951669793454025 + 0.0007936505811533442), $MachinePrecision] + -0.00277777777751721), $MachinePrecision] + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 3.5 < z

   1. Initial program 40.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} \cdot x + 1 \cdot x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 38.0%

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

   6. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, x\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, x\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \color{blue}{\frac{-1 \cdot \left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right)}{z}}, x\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \color{blue}{\frac{-1 \cdot \left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right)}{z}}, x\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}\right)\right)}}{z}, x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\mathsf{neg}\left(\color{blue}{\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000}\right)\right)\right)}\right)}{z}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000}\right)\right) + \frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}\right)}\right)}{z}, x\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}\right)\right)}}{z}, x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\left(\mathsf{neg}\left(\color{blue}{\frac{-751220861604756070699018739433}{10000000000000000000000000000000}}\right)\right) + \left(\mathsf{neg}\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}\right)\right)}{z}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}\right)\right)}{z}, x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \left(\mathsf{neg}\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}\right)\right)}}{z}, x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot 1}{z}}\right)\right)}{z}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}}{z}\right)\right)}{z}, x\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}\right)}{z}}}{z}, x\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{692910599291889}{10000000000000000} + \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}\right)}{z}}}{z}, x\right) \]

      15. metadata-eval 98.0

          \[\leadsto \mathsf{fma}\left(y, 0.0692910599291889 + \frac{0.07512208616047561 + \frac{\color{blue}{-0.4046220386999212}}{z}}{z}, x\right) \]

   8. Simplified 98.0%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{0.0692910599291889 + \frac{0.07512208616047561 + \frac{-0.4046220386999212}{z}}{z}}, x\right) \]

   if -5.4000000000000004 < z < 3.5

   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 x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} \cdot x + 1 \cdot x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, x\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right) + \frac{279195317918525}{3350343815022304}}, x\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, \frac{279195317918525}{3350343815022304}\right)}, x\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) + \left(\mathsf{neg}\left(\frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)\right)}, \frac{279195317918525}{3350343815022304}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) + \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080}}, \frac{279195317918525}{3350343815022304}\right), x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}\right)}, \frac{279195317918525}{3350343815022304}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z + \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320}}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}\right), \frac{279195317918525}{3350343815022304}\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280}} + \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}\right), \frac{279195317918525}{3350343815022304}\right), x\right) \]

      8. accelerator-lowering-fma.f64 98.6

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.0005951669793454025, 0.0007936505811533442\right)}, -0.00277777777751721\right), 0.08333333333333323\right), x\right) \]

   8. Simplified 98.6%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.0005951669793454025, 0.0007936505811533442\right), -0.00277777777751721\right), 0.08333333333333323\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.0005951669793454025, 0.0007936505811533442\right), -0.00277777777751721\right), 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)))
   (if (<= z -5.4)
     t_0
     (if (<= z 3.0)
       (fma
        y
        (fma
         z
         (fma
          z
          (fma z -0.0005951669793454025 0.0007936505811533442)
          -0.00277777777751721)
         0.08333333333333323)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	double tmp;
	if (z <= -5.4) {
		tmp = t_0;
	} else if (z <= 3.0) {
		tmp = fma(y, fma(z, fma(z, fma(z, -0.0005951669793454025, 0.0007936505811533442), -0.00277777777751721), 0.08333333333333323), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x)
	tmp = 0.0
	if (z <= -5.4)
		tmp = t_0;
	elseif (z <= 3.0)
		tmp = fma(y, fma(z, fma(z, fma(z, -0.0005951669793454025, 0.0007936505811533442), -0.00277777777751721), 0.08333333333333323), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.4], t$95$0, If[LessEqual[z, 3.0], N[(y * N[(z * N[(z * N[(z * -0.0005951669793454025 + 0.0007936505811533442), $MachinePrecision] + -0.00277777777751721), $MachinePrecision] + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 3 < z

   1. Initial program 40.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-+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

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

   5. Simplified 97.7%

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

   if -5.4000000000000004 < z < 3

   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 x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} \cdot x + 1 \cdot x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)}\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{\frac{11167812716741}{40000000000000} + z \cdot \left(\frac{307332350656623}{625000000000000} + \frac{692910599291889}{10000000000000000} \cdot z\right)}{x \cdot \left(\frac{104698244219447}{31250000000000} + z \cdot \left(\frac{6012459259764103}{1000000000000000} + z\right)\right)} \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, x\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right) + \frac{279195317918525}{3350343815022304}}, x\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, \frac{279195317918525}{3350343815022304}\right)}, x\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) + \left(\mathsf{neg}\left(\frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)\right)}, \frac{279195317918525}{3350343815022304}\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) + \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080}}, \frac{279195317918525}{3350343815022304}\right), x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}\right)}, \frac{279195317918525}{3350343815022304}\right), x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z + \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320}}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}\right), \frac{279195317918525}{3350343815022304}\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280}} + \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}\right), \frac{279195317918525}{3350343815022304}\right), x\right) \]

      8. accelerator-lowering-fma.f64 98.6

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.0005951669793454025, 0.0007936505811533442\right)}, -0.00277777777751721\right), 0.08333333333333323\right), x\right) \]

   8. Simplified 98.6%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.0005951669793454025, 0.0007936505811533442\right), -0.00277777777751721\right), 0.08333333333333323\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, \mathsf{fma}\left(z, y \cdot -0.00277777777751721, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)))
   (if (<= z -5.4)
     t_0
     (if (<= z 5.0)
       (fma y 0.08333333333333323 (fma z (* y -0.00277777777751721) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	double tmp;
	if (z <= -5.4) {
		tmp = t_0;
	} else if (z <= 5.0) {
		tmp = fma(y, 0.08333333333333323, fma(z, (y * -0.00277777777751721), x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x)
	tmp = 0.0
	if (z <= -5.4)
		tmp = t_0;
	elseif (z <= 5.0)
		tmp = fma(y, 0.08333333333333323, fma(z, Float64(y * -0.00277777777751721), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.4], t$95$0, If[LessEqual[z, 5.0], N[(y * 0.08333333333333323 + N[(z * N[(y * -0.00277777777751721), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5 < z

   1. Initial program 40.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-+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

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

   5. Simplified 97.7%

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

   if -5.4000000000000004 < z < 5

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{279195317918525}{3350343815022304}, \mathsf{fma}\left(z, \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot y}, x\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{279195317918525}{3350343815022304}, \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{-155900051080628738716045985239}{56124018394291031809500087342080}}, x\right)\right) \]

      2. *-lowering-*.f64 98.4

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

   8. Simplified 98.4%

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

Alternative 8: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, \mathsf{fma}\left(z, y \cdot -0.00277777777751721, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y 0.0692910599291889 x)
   (if (<= z 5.0)
     (fma y 0.08333333333333323 (fma z (* y -0.00277777777751721) x))
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 5.0) {
		tmp = fma(y, 0.08333333333333323, fma(z, (y * -0.00277777777751721), x));
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 5.0)
		tmp = fma(y, 0.08333333333333323, fma(z, Float64(y * -0.00277777777751721), x));
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 5.0], N[(y * 0.08333333333333323 + N[(z * N[(y * -0.00277777777751721), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5 < z

   1. Initial program 40.1%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if -5.4000000000000004 < z < 5

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{279195317918525}{3350343815022304}, \mathsf{fma}\left(z, \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot y}, x\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{279195317918525}{3350343815022304}, \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{-155900051080628738716045985239}{56124018394291031809500087342080}}, x\right)\right) \]

      2. *-lowering-*.f64 98.4

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

   8. Simplified 98.4%

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

Alternative 9: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.00277777777751721, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y 0.0692910599291889 x)
   (if (<= z 5.0)
     (fma y (fma z -0.00277777777751721 0.08333333333333323) x)
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 5.0) {
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 5.0)
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 5.0], N[(y * N[(z * -0.00277777777751721 + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5 < z

   1. Initial program 40.1%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if -5.4000000000000004 < z < 5

   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. accelerator-lowering-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. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. metadata-eval 98.4

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

   5. Simplified 98.4%

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

Alternative 10: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 5.6:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y 0.0692910599291889 x)
   (if (<= z 5.6) (fma y 0.08333333333333323 x) (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 5.6) {
		tmp = fma(y, 0.08333333333333323, x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 5.6)
		tmp = fma(y, 0.08333333333333323, x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 5.6], N[(y * 0.08333333333333323 + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 5.5999999999999996 < z

   1. Initial program 40.1%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if -5.4000000000000004 < z < 5.5999999999999996

   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 11: 59.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.92 \cdot 10^{-49}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.92e-49)
   (* y 0.0692910599291889)
   (if (<= y 2.2e+95) x (* y 0.0692910599291889))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.92e-49) {
		tmp = y * 0.0692910599291889;
	} else if (y <= 2.2e+95) {
		tmp = x;
	} else {
		tmp = y * 0.0692910599291889;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.92d-49)) then
        tmp = y * 0.0692910599291889d0
    else if (y <= 2.2d+95) then
        tmp = x
    else
        tmp = y * 0.0692910599291889d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.92e-49) {
		tmp = y * 0.0692910599291889;
	} else if (y <= 2.2e+95) {
		tmp = x;
	} else {
		tmp = y * 0.0692910599291889;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.92e-49:
		tmp = y * 0.0692910599291889
	elif y <= 2.2e+95:
		tmp = x
	else:
		tmp = y * 0.0692910599291889
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.92e-49)
		tmp = Float64(y * 0.0692910599291889);
	elseif (y <= 2.2e+95)
		tmp = x;
	else
		tmp = Float64(y * 0.0692910599291889);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.92e-49)
		tmp = y * 0.0692910599291889;
	elseif (y <= 2.2e+95)
		tmp = x;
	else
		tmp = y * 0.0692910599291889;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.92e-49], N[(y * 0.0692910599291889), $MachinePrecision], If[LessEqual[y, 2.2e+95], x, N[(y * 0.0692910599291889), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9199999999999999e-49 or 2.1999999999999999e95 < y

   1. Initial program 61.6%

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

   3. Taylor expanded in z around 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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 63.8

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

   5. Simplified 63.8%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 47.7

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

   8. Simplified 47.7%

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

   if -3.9199999999999999e-49 < y < 2.1999999999999999e95

   1. Initial program 76.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 75.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.92 \cdot 10^{-49}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.4% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

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 x around inf

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

5. Simplified 48.1%

   \[\leadsto \color{blue}{x} \]
6. 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

real(8) function code(x, y, z)
    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 2024205 
(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))))