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

Average Error: 29.3 → 0.8

Time: 14.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{t}{z}}{z}\\ t_2 := \frac{457.9610022158428}{z \cdot z}\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(t_1 + \frac{a + -5864.8025282699045}{{z}^{3}}\right) + \left(t_2 + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{elif}\;z \leq 9.3 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right) + b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t_2 + \left(3.13060547623 + \left(\frac{\mathsf{fma}\left(t, -15.234687407, -5864.8025282699045\right)}{{z}^{3}} + \left(t_1 + \frac{-36.52704169880642}{z}\right)\right)\right), x\right)\\ \end{array} \]

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
 (let* ((t_1 (/ (/ t z) z)) (t_2 (/ 457.9610022158428 (* z z))))
   (if (<= z -8.6e+23)
     (fma
      y
      (+
       3.13060547623
       (+
        (+ t_1 (/ (+ a -5864.8025282699045) (pow z 3.0)))
        (+ t_2 (/ -36.52704169880642 z))))
      x)
     (if (<= z 9.3e+52)
       (fma
        y
        (/
         (+ (* z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a)) b)
         (fma
          z
          (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
          0.607771387771))
        x)
       (fma
        y
        (+
         t_2
         (+
          3.13060547623
          (+
           (/ (fma t -15.234687407 -5864.8025282699045) (pow z 3.0))
           (+ t_1 (/ -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 t_1 = (t / z) / z;
	double t_2 = 457.9610022158428 / (z * z);
	double tmp;
	if (z <= -8.6e+23) {
		tmp = fma(y, (3.13060547623 + ((t_1 + ((a + -5864.8025282699045) / pow(z, 3.0))) + (t_2 + (-36.52704169880642 / z)))), x);
	} else if (z <= 9.3e+52) {
		tmp = fma(y, (((z * fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a)) + b) / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x);
	} else {
		tmp = fma(y, (t_2 + (3.13060547623 + ((fma(t, -15.234687407, -5864.8025282699045) / pow(z, 3.0)) + (t_1 + (-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)
	t_1 = Float64(Float64(t / z) / z)
	t_2 = Float64(457.9610022158428 / Float64(z * z))
	tmp = 0.0
	if (z <= -8.6e+23)
		tmp = fma(y, Float64(3.13060547623 + Float64(Float64(t_1 + Float64(Float64(a + -5864.8025282699045) / (z ^ 3.0))) + Float64(t_2 + Float64(-36.52704169880642 / z)))), x);
	elseif (z <= 9.3e+52)
		tmp = fma(y, Float64(Float64(Float64(z * fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a)) + b) / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x);
	else
		tmp = fma(y, Float64(t_2 + Float64(3.13060547623 + Float64(Float64(fma(t, -15.234687407, -5864.8025282699045) / (z ^ 3.0)) + Float64(t_1 + 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_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(457.9610022158428 / N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.6e+23], N[(y * N[(3.13060547623 + N[(N[(t$95$1 + N[(N[(a + -5864.8025282699045), $MachinePrecision] / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(-36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 9.3e+52], N[(y * N[(N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(t$95$2 + N[(3.13060547623 + N[(N[(N[(t * -15.234687407 + -5864.8025282699045), $MachinePrecision] / N[Power[z, 3.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(-36.52704169880642 / z), $MachinePrecision]), $MachinePrecision]), $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	29.3
Target	1.1
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if z < -8.5999999999999997e23

   1. Initial program 58.2

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

   2. Simplified 55.4

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

   3. Taylor expanded in z around -inf 0.8

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right)\right) - 36.52704169880642 \cdot \frac{1}{z}}, x\right) \]

   4. Simplified 0.8

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

   5. Taylor expanded in t around 0 0.7

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

   6. Simplified 0.7

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

   if -8.5999999999999997e23 < z < 9.29999999999999994e52

   1. Initial program 1.5

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

   2. Simplified 0.7

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

   3. Applied egg-rr 0.7

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

   if 9.29999999999999994e52 < z

   1. Initial program 61.5

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

   2. Simplified 59.7

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

   3. Taylor expanded in z around -inf 0.5

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \left(\frac{t}{{z}^{2}} + -1 \cdot \frac{-1 \cdot a - \left(1112.0901850848957 + -15.234687407 \cdot \left(457.9610022158428 + t\right)\right)}{{z}^{3}}\right)\right)\right) - 36.52704169880642 \cdot \frac{1}{z}}, x\right) \]

   4. Simplified 0.5

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

   5. Taylor expanded in a around 0 1.0

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

   6. Simplified 1.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 + \left(\left(\frac{\frac{t}{z}}{z} + \frac{a + -5864.8025282699045}{{z}^{3}}\right) + \left(\frac{457.9610022158428}{z \cdot z} + \frac{-36.52704169880642}{z}\right)\right), x\right)\\ \mathbf{elif}\;z \leq 9.3 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right) + b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{457.9610022158428}{z \cdot z} + \left(3.13060547623 + \left(\frac{\mathsf{fma}\left(t, -15.234687407, -5864.8025282699045\right)}{{z}^{3}} + \left(\frac{\frac{t}{z}}{z} + \frac{-36.52704169880642}{z}\right)\right)\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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))))