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

Average Error: 26.8 → 1.0

Time: 29.0s

Precision: binary64

Cost: 46536

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+33}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 - \frac{\frac{-3451.550173699799}{x}}{x}\right) + \frac{y + -124074.40615218398}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+64}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(x + -2\right) \cdot t_0\right)\\ \end{array} \]

FPCore

(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 (sqrt (+ 4.16438922228 (/ -101.7851458539211 x)))))
   (if (<= x -2.05e+33)
     (*
      (+ x -2.0)
      (+
       (+
        (- 4.16438922228 (/ (/ -3451.550173699799 x) x))
        (/ (+ y -124074.40615218398) (pow x 3.0)))
       (/ -101.7851458539211 x)))
     (if (<= x 8e+64)
       (*
        (+ x -2.0)
        (/
         (fma
          x
          (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
          z)
         (fma
          x
          (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
          47.066876606)))
       (* t_0 (* (+ x -2.0) t_0))))))

C

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 = sqrt((4.16438922228 + (-101.7851458539211 / x)));
	double tmp;
	if (x <= -2.05e+33) {
		tmp = (x + -2.0) * (((4.16438922228 - ((-3451.550173699799 / x) / x)) + ((y + -124074.40615218398) / pow(x, 3.0))) + (-101.7851458539211 / x));
	} else if (x <= 8e+64) {
		tmp = (x + -2.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = t_0 * ((x + -2.0) * t_0);
	}
	return tmp;
}

Julia

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 = sqrt(Float64(4.16438922228 + Float64(-101.7851458539211 / x)))
	tmp = 0.0
	if (x <= -2.05e+33)
		tmp = Float64(Float64(x + -2.0) * Float64(Float64(Float64(4.16438922228 - Float64(Float64(-3451.550173699799 / x) / x)) + Float64(Float64(y + -124074.40615218398) / (x ^ 3.0))) + Float64(-101.7851458539211 / x)));
	elseif (x <= 8e+64)
		tmp = Float64(Float64(x + -2.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(t_0 * Float64(Float64(x + -2.0) * t_0));
	end
	return tmp
end

Wolfram

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[Sqrt[N[(4.16438922228 + N[(-101.7851458539211 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.05e+33], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(N[(4.16438922228 - N[(N[(-3451.550173699799 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(y + -124074.40615218398), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-101.7851458539211 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+64], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(x + -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	26.8
Target	0.7
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if x < -2.04999999999999997e33

   1. Initial program 59.1

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

   2. Simplified 55.0

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]
      Proof
      (*.f64 (+.f64 x -2) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 2))) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 2)) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000)) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000)) 4297481763/31250000)) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x)) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000)) y)) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x)) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y)) z)) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x)) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 x 216700011257/5000000000)) 263505074721/1000000000)) 156699607947/500000000) 23533438303/500000000))): 9 points increase in error, 8 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x 216700011257/5000000000) x)) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000)) 156699607947/500000000)) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x)) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000)) 23533438303/500000000)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000) x)) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 x 2) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000) x) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around -inf 1.6

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\left(-1 \cdot \frac{124074.40615218398 + -1 \cdot y}{{x}^{3}} + \left(4.16438922228 + 3451.550173699799 \cdot \frac{1}{{x}^{2}}\right)\right) - 101.7851458539211 \cdot \frac{1}{x}\right)} \]

   4. Simplified 1.6

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\left(\left(4.16438922228 + \frac{\frac{3451.550173699799}{x}}{x}\right) - \frac{124074.40615218398 - y}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)} \]
      Proof
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (/.f64 (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 x) x)) (/.f64 (-.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 y) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (Rewrite<= associate-/r*_binary64 (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x)))) (/.f64 (-.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 y) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 0 points increase in error, 14 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (/.f64 (Rewrite<= metadata-eval (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 1)) (*.f64 x x))) (/.f64 (-.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 y) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 14 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (/.f64 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 1) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (/.f64 (-.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 y) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (Rewrite<= associate-*r/_binary64 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))) (/.f64 (-.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 y) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 0 points increase in error, 14 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2)))) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (neg.f64 y))) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 14 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (-.f64 (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2)))) (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) (pow.f64 x 3))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 0 points increase in error, 14 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2)))) (neg.f64 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 0 points increase in error, 14 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (+.f64 (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 14 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))) (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2)))))) (/.f64 -12723143231740136880149/125000000000000000000 x))): 14 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))) (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))) (/.f64 (Rewrite<= metadata-eval (*.f64 -12723143231740136880149/125000000000000000000 1)) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))) (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))) (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 12723143231740136880149/125000000000000000000)) 1) x))): 0 points increase in error, 14 points decrease in error
      (*.f64 (+.f64 x -2) (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))) (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 12723143231740136880149/125000000000000000000) (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x -2) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 387732519225574910908939577061312055388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 3))) (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))) (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))))): 14 points increase in error, 0 points decrease in error

   if -2.04999999999999997e33 < x < 8.00000000000000017e64

   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. Simplified 0.6

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]
      Proof
      (*.f64 (+.f64 x -2) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 2))) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 2)) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000)) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000)) 4297481763/31250000)) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x)) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000)) y)) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x)) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y)) z)) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x)) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 x 216700011257/5000000000)) 263505074721/1000000000)) 156699607947/500000000) 23533438303/500000000))): 9 points increase in error, 8 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x 216700011257/5000000000) x)) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000)) 156699607947/500000000)) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x)) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000)) 23533438303/500000000)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000) x)) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 x 2) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000) x) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error

   if 8.00000000000000017e64 < x

   1. Initial program 64.0

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

   2. Simplified 61.6

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]
      Proof
      (*.f64 (+.f64 x -2) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 2))) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 2)) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (fma.f64 x 104109730557/25000000000 393497462077/5000000000) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000)) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000)) 4297481763/31250000)) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x)) 4297481763/31250000) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000)) y)) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x)) y) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y)) z)) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x)) z) (fma.f64 x (fma.f64 x (fma.f64 x (+.f64 x 216700011257/5000000000) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 x 216700011257/5000000000)) 263505074721/1000000000)) 156699607947/500000000) 23533438303/500000000))): 9 points increase in error, 8 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x 216700011257/5000000000) x)) 263505074721/1000000000) 156699607947/500000000) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000)) 156699607947/500000000)) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (fma.f64 x (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x)) 156699607947/500000000) 23533438303/500000000))): 0 points increase in error, 17 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000)) 23533438303/500000000)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 x 2) (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000) x)) 23533438303/500000000))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 x 2) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x 216700011257/5000000000) x) 263505074721/1000000000) x) 156699607947/500000000) x) 23533438303/500000000))): 17 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 1.7

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(4.16438922228 - 101.7851458539211 \cdot \frac{1}{x}\right)} \]

   4. Applied egg-rr 43.2

      \[\leadsto \color{blue}{\frac{\left(4 - x \cdot x\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)}{-2 - x}} \]

   5. Simplified 43.1

      \[\leadsto \color{blue}{\frac{4 - x \cdot x}{\frac{-2 - x}{4.16438922228 + \frac{-101.7851458539211}{x}}}} \]
      Proof
      (/.f64 (-.f64 4 (*.f64 x x)) (/.f64 (-.f64 -2 x) (+.f64 104109730557/25000000000 (/.f64 -12723143231740136880149/125000000000000000000 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 4 (*.f64 x x)) (+.f64 104109730557/25000000000 (/.f64 -12723143231740136880149/125000000000000000000 x))) (-.f64 -2 x))): 2 points increase in error, 0 points decrease in error

   6. Applied egg-rr 1.4

      \[\leadsto \color{blue}{\left(\left(x + -2\right) \cdot \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}\right) \cdot \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+33}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 - \frac{\frac{-3451.550173699799}{x}}{x}\right) + \frac{y + -124074.40615218398}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+64}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{4.16438922228 + \frac{-101.7851458539211}{x}} \cdot \left(\left(x + -2\right) \cdot \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	7940

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+32}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\left(\left(4.16438922228 - \frac{\frac{-3451.550173699799}{x}}{x}\right) + \frac{y + -124074.40615218398}{{x}^{3}}\right) + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(x \cdot \left(x + -2\right)\right) \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)}{t_0} + z \cdot \left(t_1 + -2 \cdot \frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(t_1 + -2 \cdot \frac{1}{47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	1.4
Cost	7492

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \left(-110.1139242984811 + \frac{y + -130977.50649958357}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(x \cdot \left(x + -2\right)\right) \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)}{t_0} + z \cdot \left(t_1 + -2 \cdot \frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(t_1 + -2 \cdot \frac{1}{47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	1.6
Cost	7240

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)\\ t_1 := \frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)\right)}{t_0}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(\frac{x}{t_0} + -2 \cdot \frac{1}{47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \end{array} \]

Alternative 4
Error	2.0
Cost	5064

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)\\ t_1 := \frac{x}{t_0}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(x \cdot \left(x + -2\right)\right) \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)}{t_0} + z \cdot \left(t_1 + -2 \cdot \frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 + z \cdot \left(t_1 + -2 \cdot \frac{1}{47.066876606 + x \cdot \left(313.399215894 + \left(x + 43.3400022514\right) \cdot \left(x \cdot x\right)\right)}\right)\\ \end{array} \]

Alternative 5
Error	2.0
Cost	2633

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+33} \lor \neg \left(x \leq 2.7 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	3.4
Cost	2121

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+26} \lor \neg \left(x \leq 4.5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot 137.519416416\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	6.2
Cost	1992

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(x \cdot \left(x + -2\right)\right) \cdot \left(y + x \cdot 137.519416416\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)}\\ \mathbf{elif}\;x \leq 48000000:\\ \;\;\;\;\left(2 - x\right) \cdot \left(x \cdot \left(z \cdot 0.14147091005106402 + y \cdot -0.0212463641547976\right) + z \cdot -0.0212463641547976\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} - \frac{55.572073733743466}{x \cdot x}\right)}\\ \end{array} \]

Alternative 8
Error	6.3
Cost	1736

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(x + -2\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 - x \cdot \left(-43.3400022514 - x\right)\right)\right)}\\ \mathbf{elif}\;x \leq 48000000:\\ \;\;\;\;\left(2 - x\right) \cdot \left(x \cdot \left(z \cdot 0.14147091005106402 + y \cdot -0.0212463641547976\right) + z \cdot -0.0212463641547976\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} - \frac{55.572073733743466}{x \cdot x}\right)}\\ \end{array} \]

Alternative 9
Error	15.0
Cost	1488

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \frac{z}{47.066876606 - x \cdot \left(x \cdot -263.505074721 + -313.399215894\right)}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 3800:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \frac{5.86923874282773}{x}}\\ \end{array} \]

Alternative 10
Error	15.0
Cost	1488

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \frac{z}{47.066876606 - x \cdot \left(x \cdot -263.505074721 + -313.399215894\right)}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} - \frac{55.572073733743466}{x \cdot x}\right)}\\ \end{array} \]

Alternative 11
Error	6.5
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 48000000:\\ \;\;\;\;\left(2 - x\right) \cdot \left(x \cdot \left(z \cdot 0.14147091005106402 + y \cdot -0.0212463641547976\right) + z \cdot -0.0212463641547976\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718 + \left(\frac{5.86923874282773}{x} - \frac{55.572073733743466}{x \cdot x}\right)}\\ \end{array} \]

Alternative 12
Error	14.7
Cost	1232

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \frac{z}{47.066876606 - x \cdot -313.399215894}\\ t_1 := \frac{x + -2}{0.24013125253755718 + \frac{5.86923874282773}{x}}\\ \mathbf{if}\;x \leq -36:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	15.0
Cost	1104

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \frac{z}{47.066876606}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]

Alternative 14
Error	14.9
Cost	1104

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \frac{z}{47.066876606}\\ t_1 := \frac{x + -2}{0.24013125253755718 + \frac{5.86923874282773}{x}}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	15.0
Cost	976

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \frac{z}{47.066876606}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.027:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 + -110.1139242984811\\ \end{array} \]

Alternative 16
Error	15.2
Cost	848

\[\begin{array}{l} t_0 := x \cdot 4.16438922228 + -110.1139242984811\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-175}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	15.2
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;\frac{x + -2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-173}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 + -110.1139242984811\\ \end{array} \]

Alternative 18
Error	15.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-165}:\\ \;\;\;\;-0.0424927283095952 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 19
Error	15.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-172}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-165}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 20
Error	15.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 1000:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 21
Error	62.6
Cost	192

\[x \cdot -0.02577432431960352 \]

Alternative 22
Error	62.6
Cost	192

\[x \cdot -50.89257292696055 \]

Alternative 23
Error	35.2
Cost	192

\[x \cdot 4.16438922228 \]

Reproduce

herbie shell --seed 2022343 
(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)))