Average Error: 26.4 → 0.6
Time: 6.6s
Precision: binary64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
↓
\[\begin{array}{l} t_0 := \frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\right)\\ t_1 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)\\ \mathbf{if}\;x \leq -3.3694724817171055 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5215728630517518 \cdot 10^{+60}:\\ \;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{y}{t_1}, x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right)}{t_1}, x \cdot x, \frac{z}{t_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- x 2.0)
   (+
    (*
     (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y)
     x)
    z))
  (+
   (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x)
   47.066876606)))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/ 3655.1204654076414 x)
          (-
           (fma x 4.16438922228 (/ y (* x x)))
           (+ 110.1139242984811 (/ 130977.50649958357 (* x x))))))
        (t_1
         (fma
          x
          (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
          47.066876606)))
   (if (<= x -3.3694724817171055e+42)
     t_0
     (if (<= x 2.5215728630517518e+60)
       (*
        (+ x -2.0)
        (fma
         (/ y t_1)
         x
         (fma
          (/ (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) t_1)
          (* x x)
          (/ z t_1))))
       t_0))))
double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}
↓
double code(double x, double y, double z) {
	double t_0 = (3655.1204654076414 / x) + (fma(x, 4.16438922228, (y / (x * x))) - (110.1139242984811 + (130977.50649958357 / (x * x))));
	double t_1 = fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606);
	double tmp;
	if (x <= -3.3694724817171055e+42) {
		tmp = t_0;
	} else if (x <= 2.5215728630517518e+60) {
		tmp = (x + -2.0) * fma((y / t_1), x, fma((fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416) / t_1), (x * x), (z / t_1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x, y, z)
	return Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606))
end
↓
function code(x, y, z)
	t_0 = Float64(Float64(3655.1204654076414 / x) + Float64(fma(x, 4.16438922228, Float64(y / Float64(x * x))) - Float64(110.1139242984811 + Float64(130977.50649958357 / Float64(x * x)))))
	t_1 = fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)
	tmp = 0.0
	if (x <= -3.3694724817171055e+42)
		tmp = t_0;
	elseif (x <= 2.5215728630517518e+60)
		tmp = Float64(Float64(x + -2.0) * fma(Float64(y / t_1), x, fma(Float64(fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416) / t_1), Float64(x * x), Float64(z / t_1))));
	else
		tmp = t_0;
	end
	return tmp
end

code[x_, y_, z_] := N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(x + 43.3400022514), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(3655.1204654076414 / x), $MachinePrecision] + N[(N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(110.1139242984811 + N[(130977.50649958357 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]}, If[LessEqual[x, -3.3694724817171055e+42], t$95$0, If[LessEqual[x, 2.5215728630517518e+60], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(y / t$95$1), $MachinePrecision] * x + N[(N[(N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}
↓
\begin{array}{l}
t_0 := \frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\right)\\
t_1 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)\\
\mathbf{if}\;x \leq -3.3694724817171055 \cdot 10^{+42}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 2.5215728630517518 \cdot 10^{+60}:\\
\;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{y}{t_1}, x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right)}{t_1}, x \cdot x, \frac{z}{t_1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Target

Original26.4
Target0.7
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.36947248171710553e42 or 2.521572863051752e60 < x

    1. Initial program 62.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

    2. Simplified58.0

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

    3. Taylor expanded in z around 0 58.0

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\frac{\left(\left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x + y\right) \cdot x}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606} + \frac{z}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606}\right)} \]

    4. Simplified57.3

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)} \]

    5. Taylor expanded in x around inf 1.0

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right)\right) - \left(110.1139242984811 + 130977.50649958357 \cdot \frac{1}{{x}^{2}}\right)} \]

    6. Simplified1.0

      \[\leadsto \color{blue}{\frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\right)} \]

    if -3.36947248171710553e42 < x < 2.521572863051752e60

    1. Initial program 1.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

    2. Simplified0.6

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

    3. Taylor expanded in y around 0 0.6

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\frac{y \cdot x}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606} + \left(\frac{\left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot {x}^{2}}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606} + \frac{z}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606}\right)\right)} \]

    4. Simplified0.3

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x \cdot x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3694724817171055 \cdot 10^{+42}:\\ \;\;\;\;\frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 2.5215728630517518 \cdot 10^{+60}:\\ \;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x \cdot x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022182 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))