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

Average Error: 20.1 → 0.1

Time: 1.4min

Precision: binary64

Cost: 20748

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)\\ t_1 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.1 \cdot 10^{+15}:\\ \;\;\;\;x + \begin{array}{l} \mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{-\frac{t_0}{\mathsf{fma}\left(z, 6.012459259764103 + z, 3.350343815022304\right)}}{\frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t_0}{3.350343815022304 - \left(-6.012459259764103 - z\right) \cdot z}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (fma
          z
          (fma z 0.0692910599291889 0.4917317610505968)
          0.279195317918525))
        (t_1 (+ x (/ y 14.431876219268936))))
   (if (<= z -5.5e+32)
     t_1
     (if (<= z 7.1e+15)
       (+
        x
        (if (!= y 0.0)
          (/
           (- (/ t_0 (fma z (+ 6.012459259764103 z) 3.350343815022304)))
           (/ -1.0 y))
          (/ (* y t_0) (- 3.350343815022304 (* (- -6.012459259764103 z) z)))))
       t_1))))

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 = fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525);
	double t_1 = x + (y / 14.431876219268936);
	double tmp;
	if (z <= -5.5e+32) {
		tmp = t_1;
	} else if (z <= 7.1e+15) {
		double tmp_1;
		if (y != 0.0) {
			tmp_1 = -(t_0 / fma(z, (6.012459259764103 + z), 3.350343815022304)) / (-1.0 / y);
		} else {
			tmp_1 = (y * t_0) / (3.350343815022304 - ((-6.012459259764103 - z) * z));
		}
		tmp = x + tmp_1;
	} else {
		tmp = t_1;
	}
	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 = fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525)
	t_1 = Float64(x + Float64(y / 14.431876219268936))
	tmp = 0.0
	if (z <= -5.5e+32)
		tmp = t_1;
	elseif (z <= 7.1e+15)
		tmp_1 = 0.0
		if (y != 0.0)
			tmp_1 = Float64(Float64(-Float64(t_0 / fma(z, Float64(6.012459259764103 + z), 3.350343815022304))) / Float64(-1.0 / y));
		else
			tmp_1 = Float64(Float64(y * t_0) / Float64(3.350343815022304 - Float64(Float64(-6.012459259764103 - z) * z)));
		end
		tmp = Float64(x + tmp_1);
	else
		tmp = t_1;
	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[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+32], t$95$1, If[LessEqual[z, 7.1e+15], N[(x + If[Unequal[y, 0.0], N[((-N[(t$95$0 / N[(z * N[(6.012459259764103 + z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]) / N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * t$95$0), $MachinePrecision] / N[(3.350343815022304 - N[(N[(-6.012459259764103 - z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision], t$95$1]]]]

Target

Original	20.1
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 2 regimes

2. if z < -5.49999999999999984e32 or 7.1e15 < z

   1. Initial program 42.8

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

   2. Taylor expanded in z around inf 0.2

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

   3. Applied egg-rr 0.0

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

   if -5.49999999999999984e32 < z < 7.1e15

   1. Initial program 0.4

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

   2. Applied egg-rr 0.4

      \[\leadsto x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\color{blue}{\frac{\left({z}^{3} + 217.34839618538297\right) \cdot z}{36.1496663503231 + z \cdot \left(z + -6.012459259764103\right)}} + 3.350343815022304} \]

   3. Applied egg-rr 0.2

      \[\leadsto x + \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\frac{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{3.350343815022304 + z \cdot \left(z + 6.012459259764103\right)}\\ } \end{array}} \]

   4. Simplified 0.2

      \[\leadsto x + \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\frac{3.350343815022304 - \left(-6.012459259764103 - z\right) \cdot z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{3.350343815022304 - \left(-6.012459259764103 - z\right) \cdot z}\\ } \end{array}} \]
      Proof

   5. Applied egg-rr 0.2

      \[\leadsto x + \begin{array}{l} \mathbf{if}\;y \ne 0:\\ \;\;\;\;\color{blue}{\frac{-\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, 6.012459259764103 + z, 3.350343815022304\right)}}{\frac{-1}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{3.350343815022304 - \left(-6.012459259764103 - z\right) \cdot z}\\ \end{array} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.1
Cost	8584

\[\begin{array}{l} t_0 := \frac{y}{-3.350343815022304 - z \cdot \left(6.012459259764103 + z\right)}\\ t_1 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+24}:\\ \;\;\;\;x + \left(t_0 \cdot \left(\left(-z\right) \cdot \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right)\right) + t_0 \cdot -0.279195317918525\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.3
Cost	8008

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{elif}\;z \leq 320:\\ \;\;\;\;x + \frac{\left(\mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right) \cdot z\right) \cdot y + 0.279195317918525 \cdot y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y - \frac{y \cdot -0.07512208616047561}{z}\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1608

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{elif}\;z \leq 320:\\ \;\;\;\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;x + \left(0.0692910599291889 \cdot y - \frac{y \cdot -0.07512208616047561}{z}\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	968

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -1950000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;x + \left(0.08333333333333323 \cdot y + \frac{z \cdot y}{-360.0000000337696}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.6
Cost	968

\[\begin{array}{l} t_0 := x + \left(0.0692910599291889 \cdot y - \frac{y \cdot -0.07512208616047561}{z}\right)\\ \mathbf{if}\;z \leq -1950000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;x + \left(0.08333333333333323 \cdot y + \frac{z \cdot y}{-360.0000000337696}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	24.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-83}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-80}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;x \leq 82:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+31}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	24.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-80}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{14.431876219268936}\\ \mathbf{elif}\;x \leq 280:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+31}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	0.8
Cost	584

\[\begin{array}{l} t_0 := x + 0.0692910599291889 \cdot y\\ \mathbf{if}\;z \leq -1950000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.8:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	0.8
Cost	584

\[\begin{array}{l} t_0 := x + 0.0692910599291889 \cdot y\\ \mathbf{if}\;z \leq -1950000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.4:\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	0.7
Cost	584

\[\begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -1950000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.6:\\ \;\;\;\;x + \frac{y}{12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	13.0
Cost	320

\[x + 0.0692910599291889 \cdot y \]

Alternative 12
Error	31.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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))))