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

Average Error: 20.0 → 0.5

Time: 10.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -105000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) + 0.4046220386999212 \cdot \frac{-1}{{z}^{2}}, x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(0.08333333333333323 + {z}^{2} \cdot 0.0007936505811533442\right) + \left(z \cdot -0.00277777777751721 + {z}^{3} \cdot -0.0005951669793454025\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.07512208616047561 \cdot \frac{y}{z} + \left(x + y \cdot 0.0692910599291889\right)\right) + \frac{y}{{z}^{2}} \cdot -0.4046220386999212\\ \end{array} \]

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 -105000.0)
   (fma
    y
    (+
     (+ 0.0692910599291889 (* 0.07512208616047561 (/ 1.0 z)))
     (* 0.4046220386999212 (/ -1.0 (pow z 2.0))))
    x)
   (if (<= z 8.2e-5)
     (fma
      y
      (+
       (+ 0.08333333333333323 (* (pow z 2.0) 0.0007936505811533442))
       (+ (* z -0.00277777777751721) (* (pow z 3.0) -0.0005951669793454025)))
      x)
     (+
      (+ (* 0.07512208616047561 (/ y z)) (+ x (* y 0.0692910599291889)))
      (* (/ y (pow z 2.0)) -0.4046220386999212)))))

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 <= -105000.0) {
		tmp = fma(y, ((0.0692910599291889 + (0.07512208616047561 * (1.0 / z))) + (0.4046220386999212 * (-1.0 / pow(z, 2.0)))), x);
	} else if (z <= 8.2e-5) {
		tmp = fma(y, ((0.08333333333333323 + (pow(z, 2.0) * 0.0007936505811533442)) + ((z * -0.00277777777751721) + (pow(z, 3.0) * -0.0005951669793454025))), x);
	} else {
		tmp = ((0.07512208616047561 * (y / z)) + (x + (y * 0.0692910599291889))) + ((y / pow(z, 2.0)) * -0.4046220386999212);
	}
	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 <= -105000.0)
		tmp = fma(y, Float64(Float64(0.0692910599291889 + Float64(0.07512208616047561 * Float64(1.0 / z))) + Float64(0.4046220386999212 * Float64(-1.0 / (z ^ 2.0)))), x);
	elseif (z <= 8.2e-5)
		tmp = fma(y, Float64(Float64(0.08333333333333323 + Float64((z ^ 2.0) * 0.0007936505811533442)) + Float64(Float64(z * -0.00277777777751721) + Float64((z ^ 3.0) * -0.0005951669793454025))), x);
	else
		tmp = Float64(Float64(Float64(0.07512208616047561 * Float64(y / z)) + Float64(x + Float64(y * 0.0692910599291889))) + Float64(Float64(y / (z ^ 2.0)) * -0.4046220386999212));
	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_] := If[LessEqual[z, -105000.0], 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] + x), $MachinePrecision], If[LessEqual[z, 8.2e-5], N[(y * N[(N[(0.08333333333333323 + N[(N[Power[z, 2.0], $MachinePrecision] * 0.0007936505811533442), $MachinePrecision]), $MachinePrecision] + N[(N[(z * -0.00277777777751721), $MachinePrecision] + N[(N[Power[z, 3.0], $MachinePrecision] * -0.0005951669793454025), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(0.07512208616047561 * N[(y / z), $MachinePrecision]), $MachinePrecision] + N[(x + N[(y * 0.0692910599291889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] * -0.4046220386999212), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.0
Target	0.4
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if z < -105000

   1. Initial program 41.3

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

   2. Simplified 33.3

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

   3. Taylor expanded in z around inf 0.3

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) - 0.4046220386999212 \cdot \frac{1}{{z}^{2}}}, x\right) \]

   if -105000 < z < 8.20000000000000009e-5

   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

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

   3. Taylor expanded in z around 0 0.4

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(0.08333333333333323 + 0.0007936505811533442 \cdot {z}^{2}\right) - \left(0.00277777777751721 \cdot z + 0.0005951669793454025 \cdot {z}^{3}\right)}, x\right) \]

   if 8.20000000000000009e-5 < z

   1. Initial program 39.2

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

   2. Simplified 31.3

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

   3. Taylor expanded in z around inf 1.0

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -105000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(0.0692910599291889 + 0.07512208616047561 \cdot \frac{1}{z}\right) + 0.4046220386999212 \cdot \frac{-1}{{z}^{2}}, x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(0.08333333333333323 + {z}^{2} \cdot 0.0007936505811533442\right) + \left(z \cdot -0.00277777777751721 + {z}^{3} \cdot -0.0005951669793454025\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.07512208616047561 \cdot \frac{y}{z} + \left(x + y \cdot 0.0692910599291889\right)\right) + \frac{y}{{z}^{2}} \cdot -0.4046220386999212\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))