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

Average Error: 19.8 → 0.1

Time: 16.5s

Precision: binary64

Cost: 26696

Math

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

↓

\[\begin{array}{l} t_0 := 14.431876219268936 + \frac{-15.646356830292042}{z}\\ \mathbf{if}\;z \leq -680000:\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + t_0}\\ \mathbf{elif}\;z \leq 255000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 14.431876219268936 (/ -15.646356830292042 z))))
   (if (<= z -680000.0)
     (+ x (/ y (+ (/ 101.23733352003822 (* z z)) t_0)))
     (if (<= z 255000000.0)
       (fma
        (/
         (fma
          z
          (fma z 0.0692910599291889 0.4917317610505968)
          0.279195317918525)
         (fma z (+ z 6.012459259764103) 3.350343815022304))
        y
        x)
       (+ x (/ y t_0))))))

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));
}

↓

double code(double x, double y, double z) {
	double t_0 = 14.431876219268936 + (-15.646356830292042 / z);
	double tmp;
	if (z <= -680000.0) {
		tmp = x + (y / ((101.23733352003822 / (z * z)) + t_0));
	} else if (z <= 255000000.0) {
		tmp = fma((fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, (z + 6.012459259764103), 3.350343815022304)), y, x);
	} else {
		tmp = x + (y / t_0);
	}
	return tmp;
}

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

↓

function code(x, y, z)
	t_0 = Float64(14.431876219268936 + Float64(-15.646356830292042 / z))
	tmp = 0.0
	if (z <= -680000.0)
		tmp = Float64(x + Float64(y / Float64(Float64(101.23733352003822 / Float64(z * z)) + t_0)));
	elseif (z <= 255000000.0)
		tmp = fma(Float64(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, Float64(z + 6.012459259764103), 3.350343815022304)), y, x);
	else
		tmp = Float64(x + Float64(y / t_0));
	end
	return tmp
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]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -680000.0], N[(x + N[(y / N[(N[(101.23733352003822 / N[(z * z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 255000000.0], N[(N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	19.8
Target	0.4
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if z < -6.8e5

   1. Initial program 40.4

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

   2. Simplified 32.1

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

      [Start] 40.4	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      associate-/l* [=>] 32.1	\[ x + \color{blue}{\frac{y}{\frac{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}}} \]
      fma-def [=>] 32.1	\[ x + \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}}{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}} \]
      fma-def [=>] 32.1	\[ x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}} \]
      fma-def [=>] 32.1	\[ x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}} \]

   3. Taylor expanded in z around inf 0.0

      \[\leadsto x + \frac{y}{\color{blue}{\left(14.431876219268936 + 101.23733352003822 \cdot \frac{1}{{z}^{2}}\right) - 15.646356830292042 \cdot \frac{1}{z}}} \]

   4. Simplified 0.0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}} \]
      Proof

      [Start] 0.0	\[ x + \frac{y}{\left(14.431876219268936 + 101.23733352003822 \cdot \frac{1}{{z}^{2}}\right) - 15.646356830292042 \cdot \frac{1}{z}} \]
      cancel-sign-sub-inv [=>] 0.0	\[ x + \frac{y}{\color{blue}{\left(14.431876219268936 + 101.23733352003822 \cdot \frac{1}{{z}^{2}}\right) + \left(-15.646356830292042\right) \cdot \frac{1}{z}}} \]
      +-commutative [=>] 0.0	\[ x + \frac{y}{\color{blue}{\left(101.23733352003822 \cdot \frac{1}{{z}^{2}} + 14.431876219268936\right)} + \left(-15.646356830292042\right) \cdot \frac{1}{z}} \]
      associate-+l+ [=>] 0.0	\[ x + \frac{y}{\color{blue}{101.23733352003822 \cdot \frac{1}{{z}^{2}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)}} \]
      associate-*r/ [=>] 0.0	\[ x + \frac{y}{\color{blue}{\frac{101.23733352003822 \cdot 1}{{z}^{2}}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)} \]
      metadata-eval [=>] 0.0	\[ x + \frac{y}{\frac{\color{blue}{101.23733352003822}}{{z}^{2}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)} \]
      unpow2 [=>] 0.0	\[ x + \frac{y}{\frac{101.23733352003822}{\color{blue}{z \cdot z}} + \left(14.431876219268936 + \left(-15.646356830292042\right) \cdot \frac{1}{z}\right)} \]
      associate-*r/ [=>] 0.0	\[ x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \color{blue}{\frac{\left(-15.646356830292042\right) \cdot 1}{z}}\right)} \]
      metadata-eval [=>] 0.0	\[ x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{\color{blue}{-15.646356830292042} \cdot 1}{z}\right)} \]
      metadata-eval [=>] 0.0	\[ x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{\color{blue}{-15.646356830292042}}{z}\right)} \]

   if -6.8e5 < z < 2.55e8

   1. Initial program 0.2

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

   2. Simplified 0.1

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

      [Start] 0.2	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      +-commutative [=>] 0.2	\[ \color{blue}{\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} + x} \]
      associate-*r/ [<=] 0.1	\[ \color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} + x \]
      *-commutative [<=] 0.1	\[ \color{blue}{\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \cdot y} + x \]
      fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(\frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right)} \]
      *-commutative [=>] 0.1	\[ \mathsf{fma}\left(\frac{\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right) \]
      fma-def [=>] 0.1	\[ \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(z, z \cdot 0.0692910599291889 + 0.4917317610505968, 0.279195317918525\right)}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right) \]
      fma-def [=>] 0.1	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}, y, x\right) \]
      *-commutative [=>] 0.1	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304}, y, x\right) \]
      fma-def [=>] 0.1	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\color{blue}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}}, y, x\right) \]

   if 2.55e8 < 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. Simplified 32.0

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

      [Start] 40.1	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      associate-/l* [=>] 32.0	\[ x + \color{blue}{\frac{y}{\frac{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}}} \]
      fma-def [=>] 32.0	\[ x + \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}}{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}} \]
      fma-def [=>] 32.0	\[ x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)}}} \]
      fma-def [=>] 32.0	\[ x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)}, z, 0.279195317918525\right)}} \]

   3. Taylor expanded in z around inf 0.1

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936 - 15.646356830292042 \cdot \frac{1}{z}}} \]

   4. Simplified 0.1

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936 - \frac{15.646356830292042}{z}}} \]
      Proof

      [Start] 0.1	\[ x + \frac{y}{14.431876219268936 - 15.646356830292042 \cdot \frac{1}{z}} \]
      associate-*r/ [=>] 0.1	\[ x + \frac{y}{14.431876219268936 - \color{blue}{\frac{15.646356830292042 \cdot 1}{z}}} \]
      metadata-eval [=>] 0.1	\[ x + \frac{y}{14.431876219268936 - \frac{\color{blue}{15.646356830292042}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -680000:\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + \left(14.431876219268936 + \frac{-15.646356830292042}{z}\right)}\\ \mathbf{elif}\;z \leq 255000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1480

\[\begin{array}{l} t_0 := 14.431876219268936 + \frac{-15.646356830292042}{z}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + t_0}\\ \mathbf{elif}\;z \leq 4.9:\\ \;\;\;\;0.0007936505811533442 \cdot \left(y \cdot \left(z \cdot z\right)\right) + \left(y \cdot 0.08333333333333323 + \left(x + -0.00277777777751721 \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_0}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1092

\[\begin{array}{l} t_0 := 14.431876219268936 + \frac{-15.646356830292042}{z}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;x + \frac{y}{\frac{101.23733352003822}{z \cdot z} + t_0}\\ \mathbf{elif}\;z \leq 6.3:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_0}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 6\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 6.3\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{+181} \lor \neg \left(z \leq 8.2 \cdot 10^{+258}\right):\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \]

Alternative 6
Error	0.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.5\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \end{array} \]

Alternative 7
Error	25.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-51}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \end{array} \]

Alternative 8
Error	25.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+20}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \end{array} \]

Alternative 9
Error	31.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023055 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< z -8120153.652456675) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x)) (if (< z 6.576118972787377e+20) (+ x (* (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))) (- (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y) (- (/ (* 0.40462203869992125 y) (* z z)) x))))

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