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

Average Error: 29.6 → 0.7

Time: 14.7s

Precision: binary64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -165000000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) + \left(\left(\frac{1}{{z}^{3}} \cdot -5864.8025282699045 + 36.52704169880642 \cdot \frac{-1}{z}\right) - 15.234687407 \cdot \frac{t}{{z}^{3}}\right), x\right)\\ \mathbf{elif}\;z \leq 130000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \left(\frac{t}{z \cdot z} + \frac{a + \mathsf{fma}\left(t, -15.234687407, -5864.8025282699045\right)}{{z}^{3}}\right)\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -165000000.0)
   (fma
    y
    (+
     (+
      3.13060547623
      (+
       (/ t (pow z 2.0))
       (+ (* 457.9610022158428 (/ 1.0 (pow z 2.0))) (/ a (pow z 3.0)))))
     (-
      (+
       (* (/ 1.0 (pow z 3.0)) -5864.8025282699045)
       (* 36.52704169880642 (/ -1.0 z)))
      (* 15.234687407 (/ t (pow z 3.0)))))
    x)
   (if (<= z 130000000000.0)
     (fma
      y
      (*
       (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
       (/
        1.0
        (fma
         z
         (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
         0.607771387771)))
      x)
     (fma
      y
      (+
       3.13060547623
       (+
        (+
         (/ 457.9610022158428 (* z z))
         (+
          (/ t (* z z))
          (/ (+ a (fma t -15.234687407 -5864.8025282699045)) (pow z 3.0))))
        (/ -36.52704169880642 z)))
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -165000000.0) {
		tmp = fma(y, ((3.13060547623 + ((t / pow(z, 2.0)) + ((457.9610022158428 * (1.0 / pow(z, 2.0))) + (a / pow(z, 3.0))))) + ((((1.0 / pow(z, 3.0)) * -5864.8025282699045) + (36.52704169880642 * (-1.0 / z))) - (15.234687407 * (t / pow(z, 3.0))))), x);
	} else if (z <= 130000000000.0) {
		tmp = fma(y, (fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) * (1.0 / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771))), x);
	} else {
		tmp = fma(y, (3.13060547623 + (((457.9610022158428 / (z * z)) + ((t / (z * z)) + ((a + fma(t, -15.234687407, -5864.8025282699045)) / pow(z, 3.0)))) + (-36.52704169880642 / z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

↓

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -165000000.0)
		tmp = fma(y, Float64(Float64(3.13060547623 + Float64(Float64(t / (z ^ 2.0)) + Float64(Float64(457.9610022158428 * Float64(1.0 / (z ^ 2.0))) + Float64(a / (z ^ 3.0))))) + Float64(Float64(Float64(Float64(1.0 / (z ^ 3.0)) * -5864.8025282699045) + Float64(36.52704169880642 * Float64(-1.0 / z))) - Float64(15.234687407 * Float64(t / (z ^ 3.0))))), x);
	elseif (z <= 130000000000.0)
		tmp = fma(y, Float64(fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) * Float64(1.0 / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771))), x);
	else
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(Float64(457.9610022158428 / Float64(z * z)) + Float64(Float64(t / Float64(z * z)) + Float64(Float64(a + fma(t, -15.234687407, -5864.8025282699045)) / (z ^ 3.0)))) + Float64(-36.52704169880642 / z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -165000000.0], N[(y * N[(N[(3.13060547623 + N[(N[(t / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(457.9610022158428 * N[(1.0 / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(1.0 / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] * -5864.8025282699045), $MachinePrecision] + N[(36.52704169880642 * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(15.234687407 * N[(t / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 130000000000.0], N[(y * N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] * N[(1.0 / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(3.13060547623 + N[(N[(N[(457.9610022158428 / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(N[(a + N[(t * -15.234687407 + -5864.8025282699045), $MachinePrecision]), $MachinePrecision] / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.6
Target	1.1
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -1.65e8

   1. Initial program 56.2

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 53.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   3. Taylor expanded in z around inf 1.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) - \left(15.234687407 \cdot \frac{t}{{z}^{3}} + \left(5864.8025282699045 \cdot \frac{1}{{z}^{3}} + 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]

   if -1.65e8 < z < 1.3e11

   1. Initial program 0.3

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   3. Applied egg-rr 0.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right) \]

   if 1.3e11 < z

   1. Initial program 57.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 54.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   3. Taylor expanded in z around inf 1.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) - \left(15.234687407 \cdot \frac{t}{{z}^{3}} + \left(5864.8025282699045 \cdot \frac{1}{{z}^{3}} + 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]

   4. Taylor expanded in t around inf 1.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) - \left(5864.8025282699045 \cdot \frac{1}{{z}^{3}} + \left(15.234687407 \cdot \frac{t}{{z}^{3}} + 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]

   5. Simplified 1.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \left(\frac{t}{z \cdot z} + \frac{a + \mathsf{fma}\left(t, -15.234687407, -5864.8025282699045\right)}{{z}^{3}}\right)\right) + \frac{-36.52704169880642}{z}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -165000000:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) + \left(\left(\frac{1}{{z}^{3}} \cdot -5864.8025282699045 + 36.52704169880642 \cdot \frac{-1}{z}\right) - 15.234687407 \cdot \frac{t}{{z}^{3}}\right), x\right)\\ \mathbf{elif}\;z \leq 130000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{457.9610022158428}{z \cdot z} + \left(\frac{t}{z \cdot z} + \frac{a + \mathsf{fma}\left(t, -15.234687407, -5864.8025282699045\right)}{{z}^{3}}\right)\right) + \frac{-36.52704169880642}{z}\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))