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

Percentage Accurate: 69.4% → 99.4%

Time: 9.8s

Alternatives: 11

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: 69.4% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.0692910599291889 \cdot y - x\\ t_1 := \mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+28} \lor \neg \left(z \leq 155000000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_0}, \left(-x\right) \cdot \frac{x}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left({t\_1}^{2} - 0.07795002554762624\right) \cdot y}{\left(t\_1 - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* 0.0692910599291889 y) x))
        (t_1 (* (fma 0.0692910599291889 z 0.4917317610505968) z)))
   (if (or (<= z -2.8e+28) (not (<= z 155000000000.0)))
     (fma (* 0.004801250986110448 y) (/ y t_0) (* (- x) (/ x t_0)))
     (+
      x
      (/
       (* (- (pow t_1 2.0) 0.07795002554762624) y)
       (*
        (- t_1 0.279195317918525)
        (fma (+ 6.012459259764103 z) z 3.350343815022304)))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0692910599291889 * y) - x;
	double t_1 = fma(0.0692910599291889, z, 0.4917317610505968) * z;
	double tmp;
	if ((z <= -2.8e+28) || !(z <= 155000000000.0)) {
		tmp = fma((0.004801250986110448 * y), (y / t_0), (-x * (x / t_0)));
	} else {
		tmp = x + (((pow(t_1, 2.0) - 0.07795002554762624) * y) / ((t_1 - 0.279195317918525) * fma((6.012459259764103 + z), z, 3.350343815022304)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0692910599291889 * y) - x)
	t_1 = Float64(fma(0.0692910599291889, z, 0.4917317610505968) * z)
	tmp = 0.0
	if ((z <= -2.8e+28) || !(z <= 155000000000.0))
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_0), Float64(Float64(-x) * Float64(x / t_0)));
	else
		tmp = Float64(x + Float64(Float64(Float64((t_1 ^ 2.0) - 0.07795002554762624) * y) / Float64(Float64(t_1 - 0.279195317918525) * fma(Float64(6.012459259764103 + z), z, 3.350343815022304))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0692910599291889 * y), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision]}, If[Or[LessEqual[z, -2.8e+28], N[Not[LessEqual[z, 155000000000.0]], $MachinePrecision]], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision] + N[((-x) * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - 0.07795002554762624), $MachinePrecision] * y), $MachinePrecision] / N[(N[(t$95$1 - 0.279195317918525), $MachinePrecision] * N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e28 or 1.55e11 < z

   1. Initial program 38.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(0.004801250986110448 \cdot y, \color{blue}{\frac{y}{0.0692910599291889 \cdot y - x}}, \left(-x\right) \cdot \frac{x}{0.0692910599291889 \cdot y - x}\right) \]

   if -2.8000000000000001e28 < z < 1.55e11

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto x + \color{blue}{\frac{\left({\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z\right)}^{2} - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+28} \lor \neg \left(z \leq 155000000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{0.0692910599291889 \cdot y - x}, \left(-x\right) \cdot \frac{x}{0.0692910599291889 \cdot y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left({\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z\right)}^{2} - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq -2 \cdot 10^{+83} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+166} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+305}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (or (<= t_0 (- INFINITY))
           (not
            (or (<= t_0 -2e+83)
                (not (or (<= t_0 5e+166) (not (<= t_0 5e+305)))))))
     (fma 0.0692910599291889 y x)
     (* 0.08333333333333323 y))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !((t_0 <= -2e+83) || !((t_0 <= 5e+166) || !(t_0 <= 5e+305)))) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = 0.08333333333333323 * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !((t_0 <= -2e+83) || !((t_0 <= 5e+166) || !(t_0 <= 5e+305))))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = Float64(0.08333333333333323 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[Or[LessEqual[t$95$0, -2e+83], N[Not[Or[LessEqual[t$95$0, 5e+166], N[Not[LessEqual[t$95$0, 5e+305]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y), $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))) < -inf.0 or -2.00000000000000006e83 < (/.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))) < 5.0000000000000002e166 or 5.00000000000000009e305 < (/.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 64.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -2.00000000000000006e83 or 5.0000000000000002e166 < (/.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))) < 5.00000000000000009e305

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.4%

      \[\leadsto 0.08333333333333323 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -\infty \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -2 \cdot 10^{+83} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+166} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+305}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+38}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* 0.0692910599291889 y)
     (if (<= t_0 -1e+38)
       (* 0.08333333333333323 y)
       (if (<= t_0 2e+146)
         (* 1.0 x)
         (if (<= t_0 5e+305)
           (* 0.08333333333333323 y)
           (* 0.0692910599291889 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0692910599291889 * y;
	} else if (t_0 <= -1e+38) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 2e+146) {
		tmp = 1.0 * x;
	} else if (t_0 <= 5e+305) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0692910599291889 * y;
	} else if (t_0 <= -1e+38) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 2e+146) {
		tmp = 1.0 * x;
	} else if (t_0 <= 5e+305) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.0692910599291889 * y
	elif t_0 <= -1e+38:
		tmp = 0.08333333333333323 * y
	elif t_0 <= 2e+146:
		tmp = 1.0 * x
	elif t_0 <= 5e+305:
		tmp = 0.08333333333333323 * y
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0692910599291889 * y);
	elseif (t_0 <= -1e+38)
		tmp = Float64(0.08333333333333323 * y);
	elseif (t_0 <= 2e+146)
		tmp = Float64(1.0 * x);
	elseif (t_0 <= 5e+305)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.0692910599291889 * y;
	elseif (t_0 <= -1e+38)
		tmp = 0.08333333333333323 * y;
	elseif (t_0 <= 2e+146)
		tmp = 1.0 * x;
	elseif (t_0 <= 5e+305)
		tmp = 0.08333333333333323 * y;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y), $MachinePrecision], If[LessEqual[t$95$0, -1e+38], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[t$95$0, 2e+146], N[(1.0 * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0 or 5.00000000000000009e305 < (/.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 1.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.4%

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

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -9.99999999999999977e37 or 1.99999999999999987e146 < (/.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))) < 5.00000000000000009e305

   1. Initial program 99.3%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. lower-fma.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.5%

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

   if -9.99999999999999977e37 < (/.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))) < 1.99999999999999987e146

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 70.4%

       \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.0692910599291889 \cdot y - x\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+28} \lor \neg \left(z \leq 720000000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_0}, \left(-x\right) \cdot \frac{x}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004801250986110448, z, 0.06814522984808499\right), z, 0.24180012482592123\right) \cdot z\right) \cdot z - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* 0.0692910599291889 y) x)))
   (if (or (<= z -2.8e+28) (not (<= z 720000000000.0)))
     (fma (* 0.004801250986110448 y) (/ y t_0) (* (- x) (/ x t_0)))
     (+
      x
      (/
       (*
        (-
         (*
          (*
           (fma
            (fma 0.004801250986110448 z 0.06814522984808499)
            z
            0.24180012482592123)
           z)
          z)
         0.07795002554762624)
        y)
       (*
        (-
         (* (fma 0.0692910599291889 z 0.4917317610505968) z)
         0.279195317918525)
        (fma (+ 6.012459259764103 z) z 3.350343815022304)))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0692910599291889 * y) - x;
	double tmp;
	if ((z <= -2.8e+28) || !(z <= 720000000000.0)) {
		tmp = fma((0.004801250986110448 * y), (y / t_0), (-x * (x / t_0)));
	} else {
		tmp = x + (((((fma(fma(0.004801250986110448, z, 0.06814522984808499), z, 0.24180012482592123) * z) * z) - 0.07795002554762624) * y) / (((fma(0.0692910599291889, z, 0.4917317610505968) * z) - 0.279195317918525) * fma((6.012459259764103 + z), z, 3.350343815022304)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0692910599291889 * y) - x)
	tmp = 0.0
	if ((z <= -2.8e+28) || !(z <= 720000000000.0))
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_0), Float64(Float64(-x) * Float64(x / t_0)));
	else
		tmp = Float64(x + Float64(Float64(Float64(Float64(Float64(fma(fma(0.004801250986110448, z, 0.06814522984808499), z, 0.24180012482592123) * z) * z) - 0.07795002554762624) * y) / Float64(Float64(Float64(fma(0.0692910599291889, z, 0.4917317610505968) * z) - 0.279195317918525) * fma(Float64(6.012459259764103 + z), z, 3.350343815022304))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0692910599291889 * y), $MachinePrecision] - x), $MachinePrecision]}, If[Or[LessEqual[z, -2.8e+28], N[Not[LessEqual[z, 720000000000.0]], $MachinePrecision]], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision] + N[((-x) * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[(N[(N[(N[(0.004801250986110448 * z + 0.06814522984808499), $MachinePrecision] * z + 0.24180012482592123), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] - 0.07795002554762624), $MachinePrecision] * y), $MachinePrecision] / N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] - 0.279195317918525), $MachinePrecision] * N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e28 or 7.2e11 < z

   1. Initial program 38.1%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.5%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(0.004801250986110448 \cdot y, \color{blue}{\frac{y}{0.0692910599291889 \cdot y - x}}, \left(-x\right) \cdot \frac{x}{0.0692910599291889 \cdot y - x}\right) \]

   if -2.8000000000000001e28 < z < 7.2e11

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto x + \color{blue}{\frac{\left({\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z\right)}^{2} - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{\color{blue}{\left({z}^{2} \cdot \left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) - \frac{124720040876201995101661081}{1600000000000000000000000000}\right)} \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left({z}^{2} \cdot \left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) - \frac{124720040876201995101661081}{1600000000000000000000000000}\right)} \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) \cdot {z}^{2}} - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      3. unpow2 N/A

         \[\leadsto x + \frac{\left(\left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) \cdot \color{blue}{\left(z \cdot z\right)} - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      4. associate-*r* N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(\left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) \cdot z\right) \cdot z} - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(\left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) \cdot z\right) \cdot z} - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(\left(\frac{94453173760125479739253764129}{390625000000000000000000000000} + z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right)\right) \cdot z\right)} \cdot z - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{\left(\left(\color{blue}{\left(z \cdot \left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right) + \frac{94453173760125479739253764129}{390625000000000000000000000000}\right)} \cdot z\right) \cdot z - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      8. *-commutative N/A

         \[\leadsto x + \frac{\left(\left(\left(\color{blue}{\left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z\right) \cdot z} + \frac{94453173760125479739253764129}{390625000000000000000000000000}\right) \cdot z\right) \cdot z - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \frac{\left(\left(\color{blue}{\mathsf{fma}\left(\frac{212953843275265618747988030847}{3125000000000000000000000000000} + \frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z, z, \frac{94453173760125479739253764129}{390625000000000000000000000000}\right)} \cdot z\right) \cdot z - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      10. +-commutative N/A

          \[\leadsto x + \frac{\left(\left(\mathsf{fma}\left(\color{blue}{\frac{480125098611044764748221188321}{100000000000000000000000000000000} \cdot z + \frac{212953843275265618747988030847}{3125000000000000000000000000000}}, z, \frac{94453173760125479739253764129}{390625000000000000000000000000}\right) \cdot z\right) \cdot z - \frac{124720040876201995101661081}{1600000000000000000000000000}\right) \cdot y}{\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right) \cdot z - \frac{11167812716741}{40000000000000}\right) \cdot \mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      11. lower-fma.f64 99.6

          \[\leadsto x + \frac{\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.004801250986110448, z, 0.06814522984808499\right)}, z, 0.24180012482592123\right) \cdot z\right) \cdot z - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto x + \frac{\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004801250986110448, z, 0.06814522984808499\right), z, 0.24180012482592123\right) \cdot z\right) \cdot z - 0.07795002554762624\right)} \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+28} \lor \neg \left(z \leq 720000000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{0.0692910599291889 \cdot y - x}, \left(-x\right) \cdot \frac{x}{0.0692910599291889 \cdot y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004801250986110448, z, 0.06814522984808499\right), z, 0.24180012482592123\right) \cdot z\right) \cdot z - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.0692910599291889 \cdot y - x\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+89} \lor \neg \left(z \leq 1100000000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{t\_0}, \left(-x\right) \cdot \frac{x}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z \cdot y, 0.279195317918525 \cdot y\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* 0.0692910599291889 y) x)))
   (if (or (<= z -9.5e+89) (not (<= z 1100000000000.0)))
     (fma (* 0.004801250986110448 y) (/ y t_0) (* (- x) (/ x t_0)))
     (+
      x
      (/
       (fma
        (fma 0.0692910599291889 z 0.4917317610505968)
        (* z y)
        (* 0.279195317918525 y))
       (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.0692910599291889 * y) - x;
	double tmp;
	if ((z <= -9.5e+89) || !(z <= 1100000000000.0)) {
		tmp = fma((0.004801250986110448 * y), (y / t_0), (-x * (x / t_0)));
	} else {
		tmp = x + (fma(fma(0.0692910599291889, z, 0.4917317610505968), (z * y), (0.279195317918525 * y)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.0692910599291889 * y) - x)
	tmp = 0.0
	if ((z <= -9.5e+89) || !(z <= 1100000000000.0))
		tmp = fma(Float64(0.004801250986110448 * y), Float64(y / t_0), Float64(Float64(-x) * Float64(x / t_0)));
	else
		tmp = Float64(x + Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), Float64(z * y), Float64(0.279195317918525 * y)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.0692910599291889 * y), $MachinePrecision] - x), $MachinePrecision]}, If[Or[LessEqual[z, -9.5e+89], N[Not[LessEqual[z, 1100000000000.0]], $MachinePrecision]], N[(N[(0.004801250986110448 * y), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision] + N[((-x) * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * N[(z * y), $MachinePrecision] + N[(0.279195317918525 * y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000003e89 or 1.1e12 < z

   1. Initial program 34.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 56.9%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, 0.004801250986110448, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y - x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.1%

      \[\leadsto \frac{\mathsf{fma}\left(0.004801250986110448 \cdot y, y, \left(-x\right) \cdot x\right)}{\color{blue}{0.0692910599291889 \cdot y} - x} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(0.004801250986110448 \cdot y, \color{blue}{\frac{y}{0.0692910599291889 \cdot y - x}}, \left(-x\right) \cdot \frac{x}{0.0692910599291889 \cdot y - x}\right) \]

   if -9.5000000000000003e89 < z < 1.1e12

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z \cdot y, 0.279195317918525 \cdot y\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+89} \lor \neg \left(z \leq 1100000000000\right):\\ \;\;\;\;\mathsf{fma}\left(0.004801250986110448 \cdot y, \frac{y}{0.0692910599291889 \cdot y - x}, \left(-x\right) \cdot \frac{x}{0.0692910599291889 \cdot y - x}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z \cdot y, 0.279195317918525 \cdot y\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15.5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{elif}\;z \leq 5.1:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, \mathsf{fma}\left(0.0692910599291889, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -15.5)
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (if (<= z 5.1)
     (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)
     (fma (/ y z) 0.07512208616047561 (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -15.5) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else if (z <= 5.1) {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	} else {
		tmp = fma((y / z), 0.07512208616047561, fma(0.0692910599291889, y, x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -15.5)
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	elseif (z <= 5.1)
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	else
		tmp = fma(Float64(y / z), 0.07512208616047561, fma(0.0692910599291889, y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -15.5], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.1], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * 0.07512208616047561 + N[(0.0692910599291889 * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -15.5

   1. Initial program 45.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if -15.5 < z < 5.0999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.0999999999999996 < z

   1. Initial program 36.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 24.8%

      \[\leadsto x + \color{blue}{\frac{\left({\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z\right)}^{2} - 0.07795002554762624\right) \cdot y}{\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right) \cdot z - 0.279195317918525\right) \cdot \mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}} \]

   5. Taylor expanded in z around -inf

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{312500000000000} \cdot y - \frac{-9083414359407179929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{\left(\frac{692910599291889}{10000000000000000} \cdot y + -1 \cdot \frac{\frac{-307332350656623}{312500000000000} \cdot y - \frac{-9083414359407179929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right)} \]

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

         \[\leadsto x + \left(\frac{692910599291889}{10000000000000000} \cdot y + -1 \cdot \frac{\color{blue}{y \cdot \left(\frac{-307332350656623}{312500000000000} - \frac{-9083414359407179929300981260567}{10000000000000000000000000000000}\right)}}{z}\right) \]

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{\frac{692910599291889}{10000000000000000} \cdot \left(y \cdot z\right) + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot y}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.4%

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

   11. Taylor expanded in z around inf

       \[\leadsto \color{blue}{\left(x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{312500000000000} \cdot \frac{y}{z}\right)\right) - \frac{9083414359407179929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}} \]
   12. Step-by-step derivation

       1. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{307332350656623}{312500000000000} \cdot \frac{y}{z}\right)} - \frac{9083414359407179929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z} \]

       2. associate--l+ N/A

          \[\leadsto \color{blue}{\left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \left(\frac{307332350656623}{312500000000000} \cdot \frac{y}{z} - \frac{9083414359407179929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]

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

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\frac{y}{z} \cdot \left(\frac{307332350656623}{312500000000000} - \frac{9083414359407179929300981260567}{10000000000000000000000000000000}\right)} \]

       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. times-frac N/A

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

       7. metadata-eval N/A

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

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

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. distribute-neg-frac2 N/A

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

       12. mul-1-neg N/A

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

       13. associate-+r+ N/A

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

       14. +-commutative N/A

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

       15. +-commutative N/A

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

       16. associate-+l+ N/A

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

   13. Applied rewrites 99.2%

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

Alternative 7: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15.5 \lor \neg \left(z \leq 5.1\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -15.5) (not (<= z 5.1)))
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -15.5) || !(z <= 5.1)) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -15.5) || !(z <= 5.1))
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -15.5], N[Not[LessEqual[z, 5.1]], $MachinePrecision]], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -15.5 or 5.0999999999999996 < 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--l+ N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

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

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if -15.5 < z < 5.0999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -15.5 \lor \neg \left(z \leq 5.1\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15.5 \lor \neg \left(z \leq 5.1\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -15.5) (not (<= z 5.1)))
   (fma 0.0692910599291889 y x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -15.5) || !(z <= 5.1)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -15.5) || !(z <= 5.1))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -15.5], N[Not[LessEqual[z, 5.1]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -15.5 or 5.0999999999999996 < 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. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -15.5 < z < 5.0999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -15.5 \lor \neg \left(z \leq 5.1\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -15.5 \lor \neg \left(z \leq 6.2\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -15.5) (not (<= z 6.2)))
   (fma 0.0692910599291889 y x)
   (fma 0.08333333333333323 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -15.5) || !(z <= 6.2)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(0.08333333333333323, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -15.5) || !(z <= 6.2))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(0.08333333333333323, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -15.5], N[Not[LessEqual[z, 6.2]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -15.5 or 6.20000000000000018 < 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. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -15.5 < z < 6.20000000000000018

   1. Initial program 99.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -15.5 \lor \neg \left(z \leq 6.2\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+72} \lor \neg \left(y \leq 10^{+112}\right):\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.5e+72) (not (<= y 1e+112)))
   (* 0.0692910599291889 y)
   (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+72) || !(y <= 1e+112)) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-5.5d+72)) .or. (.not. (y <= 1d+112))) then
        tmp = 0.0692910599291889d0 * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+72) || !(y <= 1e+112)) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.5e+72) or not (y <= 1e+112):
		tmp = 0.0692910599291889 * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.5e+72) || !(y <= 1e+112))
		tmp = Float64(0.0692910599291889 * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.5e+72) || ~((y <= 1e+112)))
		tmp = 0.0692910599291889 * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.5e+72], N[Not[LessEqual[y, 1e+112]], $MachinePrecision]], N[(0.0692910599291889 * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e72 or 9.9999999999999993e111 < y

   1. Initial program 64.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.5%

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

   if -5.5e72 < y < 9.9999999999999993e111

   1. Initial program 75.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.0%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+72} \lor \neg \left(y \leq 10^{+112}\right):\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0692910599291889d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 75.7

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.5%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024320 
(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))))