Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.7 → 8.9

Time: 24.8s

Precision: binary64

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)\\ t_2 := \mathsf{fma}\left(y, t_1, i\right)\\ t_3 := \frac{t}{t_2}\\ t_4 := \frac{z}{y} + \left(x - x \cdot \frac{a}{y}\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+84}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+17}:\\ \;\;\;\;t_3 + \left(x \cdot \left({y}^{3} \cdot \frac{1}{t_1}\right) + \left(\frac{y}{\frac{t_2}{\mathsf{fma}\left(y, 27464.7644705, 230661.510616\right)}} + \frac{{y}^{3}}{\frac{t_2}{z}}\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+39}:\\ \;\;\;\;t_3 + \left(y \cdot \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(x, y \cdot y, 27464.7644705\right), 230661.510616\right)}{t_2} + \frac{z \cdot {y}^{3}}{t_2}\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{t}{y}}{t_1} + \left(x \cdot \frac{{y}^{4}}{t_2} + \frac{y}{\frac{t_2}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (+ y a) b) c))
        (t_2 (fma y t_1 i))
        (t_3 (/ t t_2))
        (t_4 (+ (/ z y) (- x (* x (/ a y))))))
   (if (<= y -2.6e+84)
     t_4
     (if (<= y -9.2e+17)
       (+
        t_3
        (+
         (* x (* (pow y 3.0) (/ 1.0 t_1)))
         (+
          (/ y (/ t_2 (fma y 27464.7644705 230661.510616)))
          (/ (pow y 3.0) (/ t_2 z)))))
       (if (<= y 1.45e+39)
         (+
          t_3
          (+
           (* y (/ (fma y (fma x (* y y) 27464.7644705) 230661.510616) t_2))
           (/ (* z (pow y 3.0)) t_2)))
         (if (<= y 2.4e+72)
           (+
            (/ (/ t y) t_1)
            (+
             (* x (/ (pow y 4.0) t_2))
             (/ y (/ t_2 (fma y (fma y z 27464.7644705) 230661.510616)))))
           t_4))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, (y + a), b), c);
	double t_2 = fma(y, t_1, i);
	double t_3 = t / t_2;
	double t_4 = (z / y) + (x - (x * (a / y)));
	double tmp;
	if (y <= -2.6e+84) {
		tmp = t_4;
	} else if (y <= -9.2e+17) {
		tmp = t_3 + ((x * (pow(y, 3.0) * (1.0 / t_1))) + ((y / (t_2 / fma(y, 27464.7644705, 230661.510616))) + (pow(y, 3.0) / (t_2 / z))));
	} else if (y <= 1.45e+39) {
		tmp = t_3 + ((y * (fma(y, fma(x, (y * y), 27464.7644705), 230661.510616) / t_2)) + ((z * pow(y, 3.0)) / t_2));
	} else if (y <= 2.4e+72) {
		tmp = ((t / y) / t_1) + ((x * (pow(y, 4.0) / t_2)) + (y / (t_2 / fma(y, fma(y, z, 27464.7644705), 230661.510616))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

↓

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, Float64(y + a), b), c)
	t_2 = fma(y, t_1, i)
	t_3 = Float64(t / t_2)
	t_4 = Float64(Float64(z / y) + Float64(x - Float64(x * Float64(a / y))))
	tmp = 0.0
	if (y <= -2.6e+84)
		tmp = t_4;
	elseif (y <= -9.2e+17)
		tmp = Float64(t_3 + Float64(Float64(x * Float64((y ^ 3.0) * Float64(1.0 / t_1))) + Float64(Float64(y / Float64(t_2 / fma(y, 27464.7644705, 230661.510616))) + Float64((y ^ 3.0) / Float64(t_2 / z)))));
	elseif (y <= 1.45e+39)
		tmp = Float64(t_3 + Float64(Float64(y * Float64(fma(y, fma(x, Float64(y * y), 27464.7644705), 230661.510616) / t_2)) + Float64(Float64(z * (y ^ 3.0)) / t_2)));
	elseif (y <= 2.4e+72)
		tmp = Float64(Float64(Float64(t / y) / t_1) + Float64(Float64(x * Float64((y ^ 4.0) / t_2)) + Float64(y / Float64(t_2 / fma(y, fma(y, z, 27464.7644705), 230661.510616)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(y * t$95$1 + i), $MachinePrecision]}, Block[{t$95$3 = N[(t / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z / y), $MachinePrecision] + N[(x - N[(x * N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+84], t$95$4, If[LessEqual[y, -9.2e+17], N[(t$95$3 + N[(N[(x * N[(N[Power[y, 3.0], $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(t$95$2 / N[(y * 27464.7644705 + 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[y, 3.0], $MachinePrecision] / N[(t$95$2 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+39], N[(t$95$3 + N[(N[(y * N[(N[(y * N[(x * N[(y * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+72], N[(N[(N[(t / y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(x * N[(N[Power[y, 4.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t$95$2 / N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

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

Bits error versus c

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if y < -2.6000000000000001e84 or 2.4000000000000001e72 < y

   1. Initial program 63.6

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf 19.1

      \[\leadsto \color{blue}{\left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y}} \]

   3. Simplified 15.1

      \[\leadsto \color{blue}{\frac{z}{y} + \left(x - \frac{a}{y} \cdot x\right)} \]

   if -2.6000000000000001e84 < y < -9.2e17

   1. Initial program 42.0

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in x around 0 42.1

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\frac{{y}^{4} \cdot x}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y \cdot \left(230661.510616 + y \cdot \left(y \cdot z + 27464.7644705\right)\right)}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}\right)} \]

   3. Simplified 33.3

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right)} \]

   4. Taylor expanded in i around 0 28.8

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\color{blue}{\frac{{y}^{3}}{c + y \cdot \left(\left(y + a\right) \cdot y + b\right)}} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right) \]

   5. Simplified 28.8

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\color{blue}{\frac{{y}^{3}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)}} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right) \]

   6. Applied egg-rr 28.8

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\color{blue}{\left({y}^{3} \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)}\right)} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right) \]

   7. Taylor expanded in z around 0 31.6

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\left({y}^{3} \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)}\right) \cdot x + \color{blue}{\left(\frac{y \cdot \left(230661.510616 + 27464.7644705 \cdot y\right)}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{{y}^{3} \cdot z}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}\right)}\right) \]

   8. Simplified 22.5

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\left({y}^{3} \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)}\right) \cdot x + \color{blue}{\left(\frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, 27464.7644705, 230661.510616\right)}} + \frac{{y}^{3}}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{z}}\right)}\right) \]

   if -9.2e17 < y < 1.45000000000000015e39

   1. Initial program 1.7

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in z around 0 1.8

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\frac{\left(230661.510616 + y \cdot \left(27464.7644705 + {y}^{2} \cdot x\right)\right) \cdot y}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{{y}^{3} \cdot z}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}\right)} \]

   3. Simplified 1.8

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(x, y \cdot y, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \cdot y + \frac{z \cdot {y}^{3}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)} \]

   if 1.45000000000000015e39 < y < 2.4000000000000001e72

   1. Initial program 46.9

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in x around 0 46.9

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \left(\frac{{y}^{4} \cdot x}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{y \cdot \left(230661.510616 + y \cdot \left(y \cdot z + 27464.7644705\right)\right)}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}\right)} \]

   3. Simplified 33.7

      \[\leadsto \color{blue}{\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right)} \]

   4. Taylor expanded in i around 0 34.2

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right)}} + \left(\frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right) \]

   5. Simplified 33.3

      \[\leadsto \color{blue}{\frac{\frac{t}{y}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)}} + \left(\frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \cdot x + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{z}{y} + \left(x - x \cdot \frac{a}{y}\right)\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(x \cdot \left({y}^{3} \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)}\right) + \left(\frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, 27464.7644705, 230661.510616\right)}} + \frac{{y}^{3}}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{z}}\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+39}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(y \cdot \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(x, y \cdot y, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \frac{z \cdot {y}^{3}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{t}{y}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right)} + \left(x \cdot \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - x \cdot \frac{a}{y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))