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

Average Error: 20.5 → 0.2

Time: 23.6s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+28}:\\ \;\;\;\;x + \frac{y}{57.727504877075745} \cdot 4\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + \frac{-y}{\frac{-1}{\frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) - 0.4046220386999212 \cdot \frac{1}{{z}^{2}}\right)\\ \end{array} \]

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
 (if (<= z -2.1e+28)
   (+ x (* (/ y 57.727504877075745) 4.0))
   (if (<= z 2.85)
     (+
      x
      (/
       (- y)
       (/
        -1.0
        (/
         (+
          (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
          0.279195317918525)
         (+ (* z (+ z 6.012459259764103)) 3.350343815022304)))))
     (+
      x
      (*
       y
       (-
        (+ 0.0692910599291889 (* 0.07512208616047561 (/ 1.0 z)))
        (* 0.4046220386999212 (/ 1.0 (pow z 2.0)))))))))

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 tmp;
	if (z <= -2.1e+28) {
		tmp = x + ((y / 57.727504877075745) * 4.0);
	} else if (z <= 2.85) {
		tmp = x + (-y / (-1.0 / (((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525) / ((z * (z + 6.012459259764103)) + 3.350343815022304))));
	} else {
		tmp = x + (y * ((0.0692910599291889 + (0.07512208616047561 * (1.0 / z))) - (0.4046220386999212 * (1.0 / pow(z, 2.0)))));
	}
	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
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

↓

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 (z <= (-2.1d+28)) then
        tmp = x + ((y / 57.727504877075745d0) * 4.0d0)
    else if (z <= 2.85d0) then
        tmp = x + (-y / ((-1.0d0) / (((z * ((z * 0.0692910599291889d0) + 0.4917317610505968d0)) + 0.279195317918525d0) / ((z * (z + 6.012459259764103d0)) + 3.350343815022304d0))))
    else
        tmp = x + (y * ((0.0692910599291889d0 + (0.07512208616047561d0 * (1.0d0 / z))) - (0.4046220386999212d0 * (1.0d0 / (z ** 2.0d0)))))
    end if
    code = tmp
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));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+28) {
		tmp = x + ((y / 57.727504877075745) * 4.0);
	} else if (z <= 2.85) {
		tmp = x + (-y / (-1.0 / (((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525) / ((z * (z + 6.012459259764103)) + 3.350343815022304))));
	} else {
		tmp = x + (y * ((0.0692910599291889 + (0.07512208616047561 * (1.0 / z))) - (0.4046220386999212 * (1.0 / Math.pow(z, 2.0)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

↓

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+28:
		tmp = x + ((y / 57.727504877075745) * 4.0)
	elif z <= 2.85:
		tmp = x + (-y / (-1.0 / (((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525) / ((z * (z + 6.012459259764103)) + 3.350343815022304))))
	else:
		tmp = x + (y * ((0.0692910599291889 + (0.07512208616047561 * (1.0 / z))) - (0.4046220386999212 * (1.0 / math.pow(z, 2.0)))))
	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)
	tmp = 0.0
	if (z <= -2.1e+28)
		tmp = Float64(x + Float64(Float64(y / 57.727504877075745) * 4.0));
	elseif (z <= 2.85)
		tmp = Float64(x + Float64(Float64(-y) / Float64(-1.0 / Float64(Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)))));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(0.0692910599291889 + Float64(0.07512208616047561 * Float64(1.0 / z))) - Float64(0.4046220386999212 * Float64(1.0 / (z ^ 2.0))))));
	end
	return tmp
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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e+28)
		tmp = x + ((y / 57.727504877075745) * 4.0);
	elseif (z <= 2.85)
		tmp = x + (-y / (-1.0 / (((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525) / ((z * (z + 6.012459259764103)) + 3.350343815022304))));
	else
		tmp = x + (y * ((0.0692910599291889 + (0.07512208616047561 * (1.0 / z))) - (0.4046220386999212 * (1.0 / (z ^ 2.0)))));
	end
	tmp_2 = 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_] := If[LessEqual[z, -2.1e+28], N[(x + N[(N[(y / 57.727504877075745), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.85], N[(x + N[((-y) / N[(-1.0 / N[(N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(0.0692910599291889 + N[(0.07512208616047561 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.4046220386999212 * N[(1.0 / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	20.5
Target	0.4
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if z < -2.09999999999999989e28

   1. Initial program 43.6

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

   2. Simplified 35.1

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

      [Start] 43.6	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-2 [=>] 43.6	\[ x + \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-49 [=>] 35.1	\[ x + \color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]
      rational.json-simplify-2 [=>] 35.1	\[ x + y \cdot \frac{\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-2 [=>] 35.1	\[ x + y \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304} \]

   3. Applied egg-rr 35.1

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

   4. Taylor expanded in z around inf 0.3

      \[\leadsto x + \frac{\color{blue}{0.0692910599291889}}{\frac{1}{y}} \]

   5. Applied egg-rr 0.1

      \[\leadsto x + \color{blue}{\frac{y}{57.727504877075745} \cdot 4} \]

   if -2.09999999999999989e28 < z < 2.85000000000000009

   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 0.2

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

      [Start] 0.3	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-49 [=>] 0.2	\[ x + \color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]
      rational.json-simplify-2 [=>] 0.2	\[ x + \left(\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525\right) \cdot \frac{y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-2 [=>] 0.2	\[ x + \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right) \cdot \frac{y}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304} \]

   3. Applied egg-rr 0.1

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

   if 2.85000000000000009 < z

   1. Initial program 40.8

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

   2. Simplified 32.0

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

      [Start] 40.8	\[ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-2 [=>] 40.8	\[ x + \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-49 [=>] 32.0	\[ x + \color{blue}{y \cdot \frac{\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}} \]
      rational.json-simplify-2 [=>] 32.0	\[ x + y \cdot \frac{\color{blue}{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right)} + 0.279195317918525}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
      rational.json-simplify-2 [=>] 32.0	\[ x + y \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{\color{blue}{z \cdot \left(z + 6.012459259764103\right)} + 3.350343815022304} \]

   3. Taylor expanded in z around inf 0.5

      \[\leadsto x + y \cdot \color{blue}{\left(\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) - 0.4046220386999212 \cdot \frac{1}{{z}^{2}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+28}:\\ \;\;\;\;x + \frac{y}{57.727504877075745} \cdot 4\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + \frac{-y}{\frac{-1}{\frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) - 0.4046220386999212 \cdot \frac{1}{{z}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1800

\[\begin{array}{l} t_0 := x + \frac{y}{57.727504877075745} \cdot 4\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+16}:\\ \;\;\;\;x + \frac{-y}{\frac{-1}{\frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1608

\[\begin{array}{l} t_0 := x + \frac{y}{57.727504877075745} \cdot 4\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{+16}:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	840

\[\begin{array}{l} t_0 := x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{if}\;z \leq -72000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + \frac{-y}{-12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.6
Cost	840

\[\begin{array}{l} t_0 := x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{if}\;z \leq -72000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + y \cdot \left(0.08333333333333323 + -0.00277777777751721 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.6
Cost	840

\[\begin{array}{l} t_0 := x + y \cdot \left(0.0692910599291889 + \frac{0.07512208616047561}{z}\right)\\ \mathbf{if}\;z \leq -72000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + \frac{y}{12.000000000000014 - -0.39999999996247915 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.7
Cost	712

\[\begin{array}{l} t_0 := x + \frac{y}{57.727504877075745} \cdot 4\\ \mathbf{if}\;z \leq -72000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + \frac{-y}{-12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.8
Cost	648

\[\begin{array}{l} t_0 := x + y \cdot 0.0692910599291889\\ \mathbf{if}\;z \leq -72000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + \frac{-y}{-12.000000000000014}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.8
Cost	584

\[\begin{array}{l} t_0 := x + y \cdot 0.0692910599291889\\ \mathbf{if}\;z \leq -72000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85:\\ \;\;\;\;x + y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	25.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-144}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-110}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	13.2
Cost	320

\[x + y \cdot 0.0692910599291889 \]

Alternative 11
Error	31.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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))))