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

Average Error: 19.8 → 0.3

Time: 12.3s

Precision: binary64

Cost: 26696

Math

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

↓

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (+ x (/ y 14.431876219268936))))
   (if (<= z -1.15e+24)
     t_0
     (if (<= z 0.038)
       (fma
        (/
         (fma
          z
          (fma z 0.0692910599291889 0.4917317610505968)
          0.279195317918525)
         (fma z (+ z 6.012459259764103) 3.350343815022304))
        y
        x)
       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 = x + (y / 14.431876219268936);
	double tmp;
	if (z <= -1.15e+24) {
		tmp = t_0;
	} else if (z <= 0.038) {
		tmp = fma((fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, (z + 6.012459259764103), 3.350343815022304)), y, x);
	} else {
		tmp = 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(x + Float64(y / 14.431876219268936))
	tmp = 0.0
	if (z <= -1.15e+24)
		tmp = t_0;
	elseif (z <= 0.038)
		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 = 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[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e+24], t$95$0, If[LessEqual[z, 0.038], 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], t$95$0]]]

Target

Original	19.8
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if z < -1.15e24 or 0.0379999999999999991 < z

   1. Initial program 41.1

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

   2. Simplified 33.2

      \[\leadsto \color{blue}{x + \frac{y}{\frac{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), z, 0.279195317918525\right)}}} \]
      Proof
      (+.f64 x (/.f64 y (/.f64 (fma.f64 (+.f64 z 6012459259764103/1000000000000000) z 104698244219447/31250000000000) (fma.f64 (fma.f64 z 692910599291889/10000000000000000 307332350656623/625000000000000) z 11167812716741/40000000000000)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 y (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)) (fma.f64 (fma.f64 z 692910599291889/10000000000000000 307332350656623/625000000000000) z 11167812716741/40000000000000)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 y (/.f64 (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000)) z 11167812716741/40000000000000)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 y (/.f64 (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000)) (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)))): 14 points increase in error, 26 points decrease in error

   3. Taylor expanded in z around inf 0.5

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936}} \]

   if -1.15e24 < z < 0.0379999999999999991

   1. Initial program 0.3

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

   2. 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
      (fma.f64 (/.f64 (fma.f64 z (fma.f64 z 692910599291889/10000000000000000 307332350656623/625000000000000) 11167812716741/40000000000000) (fma.f64 z (+.f64 z 6012459259764103/1000000000000000) 104698244219447/31250000000000)) y x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (fma.f64 z (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000)) 11167812716741/40000000000000) (fma.f64 z (+.f64 z 6012459259764103/1000000000000000) 104698244219447/31250000000000)) y x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000)) 11167812716741/40000000000000)) (fma.f64 z (+.f64 z 6012459259764103/1000000000000000) 104698244219447/31250000000000)) y x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z)) 11167812716741/40000000000000) (fma.f64 z (+.f64 z 6012459259764103/1000000000000000) 104698244219447/31250000000000)) y x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (+.f64 z 6012459259764103/1000000000000000)) 104698244219447/31250000000000))) y x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z)) 104698244219447/31250000000000)) y x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000) (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)) y) x)): 3 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000) y) (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000))) x): 31 points increase in error, 8 points decrease in error
      (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000))) (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000)) (+.f64 (*.f64 (+.f64 z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	20424

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right) \cdot \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	14408

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;x + \frac{y \cdot \left(0.279195317918525 + z \cdot \left(0.4917317610505968 + z \cdot 0.0692910599291889\right)\right)}{3.350343815022304 + \mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \left(z + 6.012459259764103\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	32.1
Cost	2304

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+277}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+260}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{+224}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{+143}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.4:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-55}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-169}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-225}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-288}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-241}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+25}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+74}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+282}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	32.1
Cost	2304

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+277}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+260}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{+226}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{+144}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -6.4:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{12.000000000000014}\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-169}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{y}{12.000000000000014}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-289}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-244}:\\ \;\;\;\;\frac{y}{12.000000000000014}\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-98}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{12.000000000000014}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+25}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+71}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+202}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+282}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	13.7
Cost	980

\[\begin{array}{l} t_0 := x + \frac{y}{12.000000000000014}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+274}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq 6600000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+67}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+278}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{\frac{101.23733352003822}{z} + -15.646356830292042}{z}}\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \end{array} \]

Alternative 7
Error	0.5
Cost	840

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;x + \frac{y}{z \cdot 0.39999999996247915 + 12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \end{array} \]

Alternative 9
Error	0.6
Cost	584

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.038:\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	24.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \end{array} \]

Alternative 11
Error	31.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(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))))