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

Average Error: 26.3 → 1.3

Time: 39.6s

Precision: binary64

Cost: 46536

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}\\ \mathbf{if}\;x \leq -1.4841005858849062 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \left(\frac{4752.4581585918595}{x} + \frac{y + -207551.7024428275}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.9437702787716945 \cdot 10^{+65}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x + -2\right) \cdot t_0, t_0, \frac{\left(x + -2\right) \cdot \frac{3451.550173699799}{x}}{x}\right)\\ \end{array} \]

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 -1.4841005858849062e+23)
     (+
      (fma x 4.16438922228 -110.1139242984811)
      (+ (/ 4752.4581585918595 x) (/ (+ y -207551.7024428275) (* x x))))
     (if (<= x 1.9437702787716945e+65)
       (*
        (+ 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)))
       (fma
        (* (+ x -2.0) t_0)
        t_0
        (/ (* (+ x -2.0) (/ 3451.550173699799 x)) x))))))

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 <= -1.4841005858849062e+23) {
		tmp = fma(x, 4.16438922228, -110.1139242984811) + ((4752.4581585918595 / x) + ((y + -207551.7024428275) / (x * x)));
	} else if (x <= 1.9437702787716945e+65) {
		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 = fma(((x + -2.0) * t_0), t_0, (((x + -2.0) * (3451.550173699799 / x)) / x));
	}
	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 <= -1.4841005858849062e+23)
		tmp = Float64(fma(x, 4.16438922228, -110.1139242984811) + Float64(Float64(4752.4581585918595 / x) + Float64(Float64(y + -207551.7024428275) / Float64(x * x))));
	elseif (x <= 1.9437702787716945e+65)
		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 = fma(Float64(Float64(x + -2.0) * t_0), t_0, Float64(Float64(Float64(x + -2.0) * Float64(3451.550173699799 / x)) / x));
	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, -1.4841005858849062e+23], N[(N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision] + N[(N[(4752.4581585918595 / x), $MachinePrecision] + N[(N[(y + -207551.7024428275), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9437702787716945e+65], 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[(N[(N[(x + -2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + N[(N[(N[(x + -2.0), $MachinePrecision] * N[(3451.550173699799 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	26.3
Target	0.9
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if x < -1.4841005858849062e23

   1. Initial program 57.0

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

   2. Taylor expanded in x around inf 57.0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\color{blue}{\left(43.3400022514 \cdot {x}^{2} + {x}^{3}\right)} + 313.399215894\right) \cdot x + 47.066876606} \]

   3. Taylor expanded in x around -inf 2.7

      \[\leadsto \color{blue}{\left(4752.4581585918595 \cdot \frac{1}{x} + \left(4.16438922228 \cdot x + -1 \cdot \frac{207551.7024428275 + -1 \cdot y}{{x}^{2}}\right)\right) - 110.1139242984811} \]

   4. Simplified 2.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \left(\frac{4752.4581585918595}{x} - \frac{207551.7024428275 - y}{x \cdot x}\right)} \]
      Proof
      (+.f64 (fma.f64 x 104109730557/25000000000 -13764240537310136880149/125000000000000000000) (-.f64 (/.f64 2970286349119912390428499159837293/625000000000000000000000000000 x) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (fma.f64 x 104109730557/25000000000 (Rewrite<= metadata-eval (neg.f64 13764240537310136880149/125000000000000000000))) (-.f64 (/.f64 2970286349119912390428499159837293/625000000000000000000000000000 x) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 104109730557/25000000000) (neg.f64 13764240537310136880149/125000000000000000000))) (-.f64 (/.f64 2970286349119912390428499159837293/625000000000000000000000000000 x) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 1 points increase in error, 2 points decrease in error
      (+.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 104109730557/25000000000 x)) (neg.f64 13764240537310136880149/125000000000000000000)) (-.f64 (/.f64 2970286349119912390428499159837293/625000000000000000000000000000 x) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x))) (-.f64 (/.f64 2970286349119912390428499159837293/625000000000000000000000000000 x) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (-.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 1)) x) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x))) (/.f64 (-.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 y) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (-.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (neg.f64 y))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (-.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (-.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (neg.f64 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (*.f64 104109730557/25000000000 x)) (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (+.f64 (*.f64 104109730557/25000000000 x) (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 104109730557/25000000000 x) (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x))) (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (*.f64 104109730557/25000000000 x))) (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 13764240537310136880149/125000000000000000000) (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (+.f64 (*.f64 104109730557/25000000000 x) (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (+.f64 (*.f64 104109730557/25000000000 x) (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2))))) (neg.f64 13764240537310136880149/125000000000000000000))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 (*.f64 2970286349119912390428499159837293/625000000000000000000000000000 (/.f64 1 x)) (+.f64 (*.f64 104109730557/25000000000 x) (*.f64 -1 (/.f64 (+.f64 648599070133835873532829296990356435388407301/3125000000000000000000000000000000000000 (*.f64 -1 y)) (pow.f64 x 2))))) 13764240537310136880149/125000000000000000000)): 0 points increase in error, 0 points decrease in error

   if -1.4841005858849062e23 < x < 1.9437702787716945e65

   1. Initial program 1.5

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]
      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))): 0 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, 0 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))): 0 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))): 0 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, 0 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))): 0 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, 0 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))): 1 points increase in error, 0 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))): 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 (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 x 216700011257/5000000000)) 263505074721/1000000000)) 156699607947/500000000) 23533438303/500000000))): 1 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 (+.f64 (Rewrite<= *-commutative_binary64 (*.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 (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, 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) (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))): 16 points increase in error, 10 points decrease in error

   if 1.9437702787716945e65 < 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 61.3

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]
      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))): 0 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, 0 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))): 0 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))): 0 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, 0 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))): 0 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, 0 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))): 1 points increase in error, 0 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))): 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 (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 x 216700011257/5000000000)) 263505074721/1000000000)) 156699607947/500000000) 23533438303/500000000))): 1 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 (+.f64 (Rewrite<= *-commutative_binary64 (*.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 (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, 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) (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))): 16 points increase in error, 10 points decrease in error

   3. Taylor expanded in x around inf 1.7

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

   4. Simplified 1.7

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\left(4.16438922228 + \frac{-101.7851458539211}{x}\right) + \frac{3451.550173699799}{x \cdot x}\right)} \]
      Proof
      (+.f64 (+.f64 104109730557/25000000000 (/.f64 -12723143231740136880149/125000000000000000000 x)) (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 104109730557/25000000000 (/.f64 (Rewrite<= metadata-eval (neg.f64 12723143231740136880149/125000000000000000000)) x)) (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 104109730557/25000000000 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 12723143231740136880149/125000000000000000000 x)))) (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 104109730557/25000000000 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 12723143231740136880149/125000000000000000000 1)) x))) (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 104109730557/25000000000 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))))) (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 104109730557/25000000000 (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x)))) (/.f64 2157218858562374472887084159837293/625000000000000000000000000000 (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 104109730557/25000000000 (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))) (/.f64 (Rewrite<= metadata-eval (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 1)) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 104109730557/25000000000 (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))) (/.f64 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 1) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 104109730557/25000000000 (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))) (Rewrite<= associate-*r/_binary64 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))): 1 points increase in error, 6 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))) (-.f64 104109730557/25000000000 (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))) 104109730557/25000000000) (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x)))): 1 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 104109730557/25000000000 (*.f64 2157218858562374472887084159837293/625000000000000000000000000000 (/.f64 1 (pow.f64 x 2))))) (*.f64 12723143231740136880149/125000000000000000000 (/.f64 1 x))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x + -2\right) \cdot \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}, \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}, \frac{\left(x + -2\right) \cdot \frac{3451.550173699799}{x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4841005858849062 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \left(\frac{4752.4581585918595}{x} + \frac{y + -207551.7024428275}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.9437702787716945 \cdot 10^{+65}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x + -2\right) \cdot \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}, \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}, \frac{\left(x + -2\right) \cdot \frac{3451.550173699799}{x}}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	20936

\[\begin{array}{l} t_0 := \sqrt{4.16438922228 + \frac{-101.7851458539211}{x}}\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)\\ \mathbf{if}\;x \leq -1.4841005858849062 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \left(\frac{4752.4581585918595}{x} + \frac{y + -207551.7024428275}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.9437702787716945 \cdot 10^{+65}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\frac{x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)}{t_1} + \frac{z}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x + -2\right) \cdot t_0, t_0, \frac{\left(x + -2\right) \cdot \frac{3451.550173699799}{x}}{x}\right)\\ \end{array} \]

Alternative 2
Error	1.5
Cost	7492

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(x \cdot \left(x + 43.3400022514\right)\right)\right)\\ \mathbf{if}\;x \leq -1.4841005858849062 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \left(\frac{4752.4581585918595}{x} + \frac{y + -207551.7024428275}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 8.218952216889351 \cdot 10^{+41}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\frac{x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)}{t_0} + \frac{z}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x}{t_1} + -2 \cdot \frac{1}{t_1}\right) + x \cdot 4.16438922228\\ \end{array} \]

Alternative 3
Error	1.5
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)}\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(x \cdot \left(x + 43.3400022514\right)\right)\right)\\ t_2 := z \cdot \left(\frac{x}{t_1} + -2 \cdot \frac{1}{t_1}\right) + x \cdot 4.16438922228\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+294}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	1.6
Cost	3656

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(x \cdot \left(x + 43.3400022514\right)\right)\right)\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)\\ t_2 := \frac{z}{t_1}\\ \mathbf{if}\;x \leq -3.4474594162454093 \cdot 10^{+37}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + t_2\right)\\ \mathbf{elif}\;x \leq 8.218952216889351 \cdot 10^{+41}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(\frac{x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)}{t_1} + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x}{t_0} + -2 \cdot \frac{1}{t_0}\right) + x \cdot 4.16438922228\\ \end{array} \]

Alternative 5
Error	2.1
Cost	2632

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)\\ t_1 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{t_0}\right)\\ \mathbf{if}\;x \leq -1.1999379325414214 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1377522586269590.8:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	3.0
Cost	2504

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)}\right)\\ \mathbf{if}\;x \leq -1.1999379325414214 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1377522586269590.8:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(78.6994924154 + x \cdot 4.16438922228\right)\right)\right)\right)}{47.066876606 - x \cdot \left(\left(x \cdot x\right) \cdot \left(-43.3400022514 - x\right) + -313.399215894\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	3.0
Cost	2376

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)\\ t_1 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{t_0}\right)\\ \mathbf{if}\;x \leq -1.4841005858849062 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1377522586269590.8:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z - x \cdot \left(x \cdot \left(x \cdot -78.6994924154 + -137.519416416\right) - y\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	3.1
Cost	2120

\[\begin{array}{l} t_0 := 47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)\\ t_1 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{t_0}\right)\\ \mathbf{if}\;x \leq -1.4841005858849062 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1377522586269590.8:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot 137.519416416\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	3.5
Cost	1992

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)}\right)\\ \mathbf{if}\;x \leq -26.591253996301774:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.7115154927638865:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z - x \cdot \left(x \cdot \left(x \cdot -78.6994924154 + -137.519416416\right) - y\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot 263.505074721\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	3.7
Cost	1736

\[\begin{array}{l} t_0 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{z}{47.066876606 + x \cdot \left(313.399215894 - x \cdot \left(x \cdot \left(-43.3400022514 - x\right) + -263.505074721\right)\right)}\right)\\ \mathbf{if}\;x \leq -26.591253996301774:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.7115154927638865:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot 137.519416416\right)\right)}{47.066876606 + x \cdot 313.399215894}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	5.1
Cost	1480

\[\begin{array}{l} \mathbf{if}\;x \leq -216.21940942499668:\\ \;\;\;\;\left(2 - x\right) \cdot \left(\left(-4.16438922228 + \frac{101.7851458539211}{x}\right) + \frac{-3451.550173699799}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 3.7115154927638865:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot \left(z + x \cdot \left(y + x \cdot 137.519416416\right)\right)}{47.066876606 + x \cdot 313.399215894}\\ \mathbf{else}:\\ \;\;\;\;-110.1139242984811 + \left(x \cdot 4.16438922228 + 3655.1204654076414 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 12
Error	15.7
Cost	1356

\[\begin{array}{l} t_0 := -110.1139242984811 + \left(x \cdot 4.16438922228 + 3655.1204654076414 \cdot \frac{1}{x}\right)\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot \left(313.399215894 + x \cdot 263.505074721\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	15.8
Cost	1356

\[\begin{array}{l} t_0 := \left(2 - x\right) \cdot \left(\left(-4.16438922228 + \frac{101.7851458539211}{x}\right) + \frac{-3451.550173699799}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot \left(313.399215894 + x \cdot 263.505074721\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	15.7
Cost	1356

\[\begin{array}{l} t_0 := \left(2 - x\right) \cdot \left(\left(-4.16438922228 + \frac{101.7851458539211}{x}\right) + \frac{-3451.550173699799}{x \cdot x}\right)\\ t_1 := 47.066876606 + x \cdot \left(313.399215894 + x \cdot 263.505074721\right)\\ \mathbf{if}\;x \leq -3525.974761826046:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(x + -2\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	6.9
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;\left(2 - x\right) \cdot \left(\left(-4.16438922228 + \frac{101.7851458539211}{x}\right) + \frac{-3451.550173699799}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;z \cdot -0.0424927283095952 - x \cdot \left(0.0212463641547976 \cdot \left(y \cdot 2 - z\right) - z \cdot 0.28294182010212804\right)\\ \mathbf{else}:\\ \;\;\;\;-110.1139242984811 + \left(x \cdot 4.16438922228 + 3655.1204654076414 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 16
Error	15.8
Cost	1100

\[\begin{array}{l} \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot 313.399215894}\\ \mathbf{else}:\\ \;\;\;\;-110.1139242984811 + x \cdot 4.16438922228\\ \end{array} \]

Alternative 17
Error	15.8
Cost	1100

\[\begin{array}{l} t_0 := -110.1139242984811 + \left(x \cdot 4.16438922228 + 3655.1204654076414 \cdot \frac{1}{x}\right)\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;\frac{\left(x + -2\right) \cdot z}{47.066876606 + x \cdot 313.399215894}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	15.7
Cost	1100

\[\begin{array}{l} t_0 := -110.1139242984811 + \left(x \cdot 4.16438922228 + 3655.1204654076414 \cdot \frac{1}{x}\right)\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{z}{47.066876606 + x \cdot 313.399215894}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	15.9
Cost	844

\[\begin{array}{l} t_0 := -110.1139242984811 + x \cdot 4.16438922228\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 + x \cdot 0.3041881842569256\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 20
Error	15.9
Cost	844

\[\begin{array}{l} t_0 := -110.1139242984811 + x \cdot 4.16438922228\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{z}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	15.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{z}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;-110.1139242984811 + x \cdot 4.16438922228\\ \end{array} \]

Alternative 22
Error	16.0
Cost	716

\[\begin{array}{l} t_0 := 4.16438922228 \cdot \left(x + -2\right)\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 4.403686162593693 \cdot 10^{-7}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 23
Error	15.9
Cost	716

\[\begin{array}{l} t_0 := -110.1139242984811 + x \cdot 4.16438922228\\ \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 3.7115154927638865:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 24
Error	16.0
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(y \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 25
Error	16.0
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -199097.98441067393:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -1.5056423881910675 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 26
Error	15.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2460801481994154 \cdot 10^{-26}:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 0.047623809736022746:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]

Alternative 27
Error	62.3
Cost	192

\[z \cdot 0.0031908184490743166 \]

Alternative 28
Error	61.9
Cost	192

\[\frac{137.519416416}{x} \]

Alternative 29
Error	35.8
Cost	192

\[x \cdot 4.16438922228 \]

Reproduce

herbie shell --seed 2022317 
(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)))