Jmat.Real.gamma, branch z greater than 0.5

Average Error: 3.9 → 2.2

Time: 41.2s

Precision: binary64

Cost: 51908

\[z > 0.5\]

Math

\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

↓

\[\begin{array}{l} t_0 := \frac{12.507343278686905}{\left(z + -1\right) + 5}\\ t_1 := \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\\ t_2 := \frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\\ t_3 := \frac{771.3234287776531}{\left(z + -1\right) + 3}\\ t_4 := \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\\ t_5 := \frac{-176.6150291621406}{\left(z + -1\right) + 4}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z + 6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(-1259.1392167224028, z \cdot 0.00147815209581367, z + 1\right)}{\left(z \cdot 0.00147815209581367\right) \cdot \left(z + 1\right)}\right) + t_3\right) + t_5\right) + t_0\right) + t_1\right) + t_4\right) + t_6\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-6.5 - z\right) + \mathsf{fma}\left(z + -0.5, \log \left(z + 6.5\right), \log \left(\sqrt{2 \cdot \pi}\right)\right)} \cdot \left(t_6 + \left(t_4 + \left(t_1 + \left(t_0 + \left(t_5 + \left(t_3 + \frac{0.9999999999994297 + {t_2}^{3}}{0.9999999999996197 + \left(t_2 \cdot t_2 + 0.9999999999998099 \cdot \left(\frac{-676.5203681218851}{z} + \frac{1259.1392167224028}{z + 1}\right)\right)}\right)\right)\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5)))
   (exp (- (+ (+ (- z 1.0) 7.0) 0.5))))
  (+
   (+
    (+
     (+
      (+
       (+
        (+
         (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0)))
         (/ -1259.1392167224028 (+ (- z 1.0) 2.0)))
        (/ 771.3234287776531 (+ (- z 1.0) 3.0)))
       (/ -176.6150291621406 (+ (- z 1.0) 4.0)))
      (/ 12.507343278686905 (+ (- z 1.0) 5.0)))
     (/ -0.13857109526572012 (+ (- z 1.0) 6.0)))
    (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0)))
   (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ 12.507343278686905 (+ (+ z -1.0) 5.0)))
        (t_1 (/ -0.13857109526572012 (+ (+ z -1.0) 6.0)))
        (t_2 (+ (/ 676.5203681218851 z) (/ -1259.1392167224028 (+ z 1.0))))
        (t_3 (/ 771.3234287776531 (+ (+ z -1.0) 3.0)))
        (t_4 (/ 9.984369578019572e-6 (+ (+ z -1.0) 7.0)))
        (t_5 (/ -176.6150291621406 (+ (+ z -1.0) 4.0)))
        (t_6 (/ 1.5056327351493116e-7 (+ (+ z -1.0) 8.0))))
   (if (<= (+ z -1.0) 142.0)
     (*
      (*
       (sqrt 2.0)
       (* (sqrt PI) (/ (exp (- -6.5 z)) (pow (+ z 6.5) (- 0.5 z)))))
      (+
       (+
        (+
         (+
          (+
           (+
            (+
             0.9999999999998099
             (/
              (fma -1259.1392167224028 (* z 0.00147815209581367) (+ z 1.0))
              (* (* z 0.00147815209581367) (+ z 1.0))))
            t_3)
           t_5)
          t_0)
         t_1)
        t_4)
       t_6))
     (*
      (exp
       (+ (- -6.5 z) (fma (+ z -0.5) (log (+ z 6.5)) (log (sqrt (* 2.0 PI))))))
      (+
       t_6
       (+
        t_4
        (+
         t_1
         (+
          t_0
          (+
           t_5
           (+
            t_3
            (/
             (+ 0.9999999999994297 (pow t_2 3.0))
             (+
              0.9999999999996197
              (+
               (* t_2 t_2)
               (*
                0.9999999999998099
                (+
                 (/ -676.5203681218851 z)
                 (/ 1259.1392167224028 (+ z 1.0)))))))))))))))))

C

double code(double z) {
	return ((sqrt((((double) M_PI) * 2.0)) * pow((((z - 1.0) + 7.0) + 0.5), ((z - 1.0) + 0.5))) * exp(-(((z - 1.0) + 7.0) + 0.5))) * ((((((((0.9999999999998099 + (676.5203681218851 / ((z - 1.0) + 1.0))) + (-1259.1392167224028 / ((z - 1.0) + 2.0))) + (771.3234287776531 / ((z - 1.0) + 3.0))) + (-176.6150291621406 / ((z - 1.0) + 4.0))) + (12.507343278686905 / ((z - 1.0) + 5.0))) + (-0.13857109526572012 / ((z - 1.0) + 6.0))) + (9.984369578019572e-6 / ((z - 1.0) + 7.0))) + (1.5056327351493116e-7 / ((z - 1.0) + 8.0)));
}

↓

double code(double z) {
	double t_0 = 12.507343278686905 / ((z + -1.0) + 5.0);
	double t_1 = -0.13857109526572012 / ((z + -1.0) + 6.0);
	double t_2 = (676.5203681218851 / z) + (-1259.1392167224028 / (z + 1.0));
	double t_3 = 771.3234287776531 / ((z + -1.0) + 3.0);
	double t_4 = 9.984369578019572e-6 / ((z + -1.0) + 7.0);
	double t_5 = -176.6150291621406 / ((z + -1.0) + 4.0);
	double t_6 = 1.5056327351493116e-7 / ((z + -1.0) + 8.0);
	double tmp;
	if ((z + -1.0) <= 142.0) {
		tmp = (sqrt(2.0) * (sqrt(((double) M_PI)) * (exp((-6.5 - z)) / pow((z + 6.5), (0.5 - z))))) * (((((((0.9999999999998099 + (fma(-1259.1392167224028, (z * 0.00147815209581367), (z + 1.0)) / ((z * 0.00147815209581367) * (z + 1.0)))) + t_3) + t_5) + t_0) + t_1) + t_4) + t_6);
	} else {
		tmp = exp(((-6.5 - z) + fma((z + -0.5), log((z + 6.5)), log(sqrt((2.0 * ((double) M_PI))))))) * (t_6 + (t_4 + (t_1 + (t_0 + (t_5 + (t_3 + ((0.9999999999994297 + pow(t_2, 3.0)) / (0.9999999999996197 + ((t_2 * t_2) + (0.9999999999998099 * ((-676.5203681218851 / z) + (1259.1392167224028 / (z + 1.0)))))))))))));
	}
	return tmp;
}

Julia

function code(z)
	return Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5) ^ Float64(Float64(z - 1.0) + 0.5))) * exp(Float64(-Float64(Float64(Float64(z - 1.0) + 7.0) + 0.5)))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(Float64(z - 1.0) + 1.0))) + Float64(-1259.1392167224028 / Float64(Float64(z - 1.0) + 2.0))) + Float64(771.3234287776531 / Float64(Float64(z - 1.0) + 3.0))) + Float64(-176.6150291621406 / Float64(Float64(z - 1.0) + 4.0))) + Float64(12.507343278686905 / Float64(Float64(z - 1.0) + 5.0))) + Float64(-0.13857109526572012 / Float64(Float64(z - 1.0) + 6.0))) + Float64(9.984369578019572e-6 / Float64(Float64(z - 1.0) + 7.0))) + Float64(1.5056327351493116e-7 / Float64(Float64(z - 1.0) + 8.0))))
end

↓

function code(z)
	t_0 = Float64(12.507343278686905 / Float64(Float64(z + -1.0) + 5.0))
	t_1 = Float64(-0.13857109526572012 / Float64(Float64(z + -1.0) + 6.0))
	t_2 = Float64(Float64(676.5203681218851 / z) + Float64(-1259.1392167224028 / Float64(z + 1.0)))
	t_3 = Float64(771.3234287776531 / Float64(Float64(z + -1.0) + 3.0))
	t_4 = Float64(9.984369578019572e-6 / Float64(Float64(z + -1.0) + 7.0))
	t_5 = Float64(-176.6150291621406 / Float64(Float64(z + -1.0) + 4.0))
	t_6 = Float64(1.5056327351493116e-7 / Float64(Float64(z + -1.0) + 8.0))
	tmp = 0.0
	if (Float64(z + -1.0) <= 142.0)
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(pi) * Float64(exp(Float64(-6.5 - z)) / (Float64(z + 6.5) ^ Float64(0.5 - z))))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(fma(-1259.1392167224028, Float64(z * 0.00147815209581367), Float64(z + 1.0)) / Float64(Float64(z * 0.00147815209581367) * Float64(z + 1.0)))) + t_3) + t_5) + t_0) + t_1) + t_4) + t_6));
	else
		tmp = Float64(exp(Float64(Float64(-6.5 - z) + fma(Float64(z + -0.5), log(Float64(z + 6.5)), log(sqrt(Float64(2.0 * pi)))))) * Float64(t_6 + Float64(t_4 + Float64(t_1 + Float64(t_0 + Float64(t_5 + Float64(t_3 + Float64(Float64(0.9999999999994297 + (t_2 ^ 3.0)) / Float64(0.9999999999996197 + Float64(Float64(t_2 * t_2) + Float64(0.9999999999998099 * Float64(Float64(-676.5203681218851 / z) + Float64(1259.1392167224028 / Float64(z + 1.0))))))))))))));
	end
	return tmp
end

Wolfram

code[z_] := N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision] + 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(N[(z - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(z - 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(z - 1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(z - 1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(z - 1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(z - 1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(z - 1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(z - 1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[z_] := Block[{t$95$0 = N[(12.507343278686905 / N[(N[(z + -1.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.13857109526572012 / N[(N[(z + -1.0), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(676.5203681218851 / z), $MachinePrecision] + N[(-1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(771.3234287776531 / N[(N[(z + -1.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(9.984369578019572e-6 / N[(N[(z + -1.0), $MachinePrecision] + 7.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-176.6150291621406 / N[(N[(z + -1.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(N[(z + -1.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z + -1.0), $MachinePrecision], 142.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Exp[N[(-6.5 - z), $MachinePrecision]], $MachinePrecision] / N[Power[N[(z + 6.5), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(-1259.1392167224028 * N[(z * 0.00147815209581367), $MachinePrecision] + N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(z * 0.00147815209581367), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(-6.5 - z), $MachinePrecision] + N[(N[(z + -0.5), $MachinePrecision] * N[Log[N[(z + 6.5), $MachinePrecision]], $MachinePrecision] + N[Log[N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$6 + N[(t$95$4 + N[(t$95$1 + N[(t$95$0 + N[(t$95$5 + N[(t$95$3 + N[(N[(0.9999999999994297 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.9999999999996197 + N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(0.9999999999998099 * N[(N[(-676.5203681218851 / z), $MachinePrecision] + N[(1259.1392167224028 / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < 142

   1. Initial program 2.3

      \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   2. Taylor expanded in z around inf 2.3

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \left({\left(6.5 + z\right)}^{\left(z - 0.5\right)} \cdot e^{-\left(6.5 + z\right)}\right)\right) \cdot \sqrt{\pi}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   3. Simplified 2.2

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z - -6.5\right)}^{\left(0.5 - z\right)}}\right)\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 -13/2 z)) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite<= unsub-neg_binary64 (+.f64 -13/2 (neg.f64 z)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 13/2)) (neg.f64 z))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 13/2 z)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 z 13/2)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 z) (neg.f64 13/2)))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z)) (neg.f64 13/2))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 z) 13/2))) (pow.f64 (-.f64 z -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 z)) -13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (Rewrite=> fma-neg_binary64 (fma.f64 1 z (neg.f64 -13/2))) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (fma.f64 1 z (Rewrite=> metadata-eval 13/2)) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1 z) 13/2)) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 (Rewrite=> *-lft-identity_binary64 z) 13/2) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (Rewrite<= +-commutative_binary64 (+.f64 13/2 z)) (-.f64 1/2 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 13/2 z) (Rewrite<= unsub-neg_binary64 (+.f64 1/2 (neg.f64 z))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 13/2 z) (+.f64 1/2 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (pow.f64 (+.f64 13/2 z) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (+.f64 13/2 z)) (+.f64 (*.f64 -1 z) 1/2))))))): 89 points increase in error, 104 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 z))))) (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))))) (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (/.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 13/2 (*.f64 -1 z)))) (+.f64 (*.f64 -1 z) 1/2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (Rewrite=> div-exp_binary64 (exp.f64 (-.f64 (-.f64 (*.f64 -1 z) 13/2) (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2))))))): 68 points increase in error, 81 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (exp.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (-.f64 (*.f64 -1 z) 13/2) (neg.f64 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (exp.f64 (+.f64 (-.f64 (*.f64 -1 z) 13/2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (Rewrite<= prod-exp_binary64 (*.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))))))): 87 points increase in error, 63 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 (PI.f64)) (Rewrite<= *-commutative_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (exp.f64 (-.f64 (*.f64 -1 z) 13/2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (exp.f64 (-.f64 (*.f64 -1 z) 13/2))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (Rewrite=> associate-*l*_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (-.f64 (*.f64 -1 z) 13/2)) (sqrt.f64 (PI.f64)))))): 34 points increase in error, 34 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 -1 z) (neg.f64 13/2)))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 z)) (neg.f64 13/2))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 z 13/2)))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (*.f64 (exp.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 13/2 z)))) (sqrt.f64 (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2))))) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64))))): 44 points increase in error, 49 points decrease in error
      (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2))) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (exp.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (+.f64 (*.f64 -1 z) 1/2))))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (exp.f64 (Rewrite=> distribute-rgt-neg-in_binary64 (*.f64 (log.f64 (-.f64 13/2 (*.f64 -1 z))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2))))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (Rewrite=> sub-neg_binary64 (+.f64 13/2 (neg.f64 (*.f64 -1 z))))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 z))))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (exp.f64 (*.f64 (log.f64 (+.f64 13/2 (Rewrite=> remove-double-neg_binary64 z))) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite=> exp-to-pow_binary64 (pow.f64 (+.f64 13/2 z) (neg.f64 (+.f64 (*.f64 -1 z) 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 110 points increase in error, 86 points decrease in error
      (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 (*.f64 -1 z)) (neg.f64 1/2)))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (+.f64 (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 z))) (neg.f64 1/2))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (+.f64 (Rewrite=> remove-double-neg_binary64 z) (neg.f64 1/2))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 (+.f64 13/2 z) (Rewrite<= sub-neg_binary64 (-.f64 z 1/2))) (sqrt.f64 2)) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) (pow.f64 (+.f64 13/2 z) (-.f64 z 1/2)))) (*.f64 (exp.f64 (neg.f64 (+.f64 13/2 z))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (sqrt.f64 2) (pow.f64 (+.f64 13/2 z) (-.f64 z 1/2))) (exp.f64 (neg.f64 (+.f64 13/2 z)))) (sqrt.f64 (PI.f64)))): 43 points increase in error, 41 points decrease in error
      (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (sqrt.f64 2) (*.f64 (pow.f64 (+.f64 13/2 z) (-.f64 z 1/2)) (exp.f64 (neg.f64 (+.f64 13/2 z)))))) (sqrt.f64 (PI.f64))): 34 points increase in error, 39 points decrease in error

   4. Applied egg-rr 2.2

      \[\leadsto \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z - -6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{1 \cdot \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   5. Applied egg-rr 2.0

      \[\leadsto \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z - -6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(1 \cdot \left(0.9999999999998099 + \color{blue}{\frac{\mathsf{fma}\left(-1259.1392167224028, z \cdot 0.00147815209581367, z + 1\right)}{\left(z + 1\right) \cdot \left(z \cdot 0.00147815209581367\right)}}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   if 142 < (-.f64 z 1)

   1. Initial program 64.0

      \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   2. Applied egg-rr 64.0

      \[\leadsto \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{0.9999999999994297 + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)}^{3}}{0.9999999999996197 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right) - 0.9999999999998099 \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right)}} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   3. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot 2} \cdot {\left(\left(z + -1\right) + 7.5\right)}^{\left(z + -0.5\right)}}{e^{\left(z + -1\right) + 7.5}}} \cdot \left(\left(\left(\left(\left(\left(\frac{0.9999999999994297 + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)}^{3}}{0.9999999999996197 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right) - 0.9999999999998099 \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]

   4. Applied egg-rr 7.7

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(z + -0.5, \log \left(z + 6.5\right), \log \left(\sqrt{\pi \cdot 2}\right)\right) - \left(z + 6.5\right)}} \cdot \left(\left(\left(\left(\left(\left(\frac{0.9999999999994297 + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)}^{3}}{0.9999999999996197 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right) - 0.9999999999998099 \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z - -1}\right)\right)} + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z - 1\right) + 8}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z + 6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(-1259.1392167224028, z \cdot 0.00147815209581367, z + 1\right)}{\left(z \cdot 0.00147815209581367\right) \cdot \left(z + 1\right)}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-6.5 - z\right) + \mathsf{fma}\left(z + -0.5, \log \left(z + 6.5\right), \log \left(\sqrt{2 \cdot \pi}\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \frac{0.9999999999994297 + {\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)}^{3}}{0.9999999999996197 + \left(\left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) \cdot \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right) + 0.9999999999998099 \cdot \left(\frac{-676.5203681218851}{z} + \frac{1259.1392167224028}{z + 1}\right)\right)}\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	43716

\[\begin{array}{l} t_0 := \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(-1259.1392167224028, z \cdot 0.00147815209581367, z + 1\right)}{\left(z \cdot 0.00147815209581367\right) \cdot \left(z + 1\right)}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \left(\left(\frac{t_0 \cdot t_0 + \frac{-594939.8317813153}{{\left(z + 2\right)}^{2}}}{t_0 + \frac{-771.3234287776531}{z + 2}} + \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 2
Error	2.2
Cost	43716

\[\begin{array}{l} t_0 := \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;z + -1 \leq 142:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot \frac{e^{-6.5 - z}}{{\left(z + 6.5\right)}^{\left(0.5 - z\right)}}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{\mathsf{fma}\left(-1259.1392167224028, z \cdot 0.00147815209581367, z + 1\right)}{\left(z \cdot 0.00147815209581367\right) \cdot \left(z + 1\right)}\right) + \frac{771.3234287776531}{\left(z + -1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z + -1\right) + 4}\right) + \frac{12.507343278686905}{\left(z + -1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z + -1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(z + -1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \left(\left(\frac{t_0 \cdot t_0 + \frac{-594939.8317813153}{{\left(z + 2\right)}^{2}}}{t_0 + \frac{-771.3234287776531}{z + 2}} + \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 3
Error	2.3
Cost	38212

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7\\ t_1 := \sqrt{2 \cdot \pi}\\ t_2 := \frac{-1259.1392167224028}{z + 1}\\ t_3 := \left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + t_2\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(t_1 \cdot {\left(0.5 + t_0\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t_0} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + t_2\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\left(\frac{t_3 \cdot t_3 + \frac{-594939.8317813153}{{\left(z + 2\right)}^{2}}}{t_3 + \frac{-771.3234287776531}{z + 2}} + \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right) \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.3
Cost	36996

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t_0} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + \frac{-1259.1392167224028}{z + 1}\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(0.5 + t_0\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(\sqrt{\pi} \cdot e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)}\right)\right) \cdot t_1\\ \end{array} \]

Alternative 5
Error	2.3
Cost	31172

\[\begin{array}{l} t_0 := \left(z + -1\right) + 7\\ t_1 := \sqrt{2 \cdot \pi}\\ t_2 := \frac{-1259.1392167224028}{z + 1}\\ \mathbf{if}\;z + -1 \leq 140:\\ \;\;\;\;\left(\left(t_1 \cdot {\left(0.5 + t_0\right)}^{\left(\left(z + -1\right) + 0.5\right)}\right) \cdot e^{-0.5 + \left(\left(1 - z\right) + -7\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(z + -1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t_0} + \left(\frac{-0.13857109526572012}{\left(z + -1\right) + 6} + \left(\frac{12.507343278686905}{\left(z + -1\right) + 5} + \left(\frac{-176.6150291621406}{\left(z + -1\right) + 4} + \left(\frac{771.3234287776531}{\left(z + -1\right) + 3} + \left(0.9999999999998099 + \left(\frac{676.5203681218851}{z} + t_2\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(t_2 + \frac{771.3234287776531}{z + 2}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	2.4
Cost	29892

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ t_1 := \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\\ \mathbf{if}\;z \leq 127.98639682064002:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{1}{\frac{1}{\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot t_1\right)\\ \end{array} \]

Alternative 7
Error	2.4
Cost	29700

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ t_1 := \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\\ \mathbf{if}\;z \leq 127.98639682064002:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(e^{\left(-6.5 - z\right) + \left(z + -0.5\right) \cdot \log \left(z + 6.5\right)} \cdot t_1\right)\\ \end{array} \]

Alternative 8
Error	3.8
Cost	29504

\[\sqrt{2 \cdot \pi} \cdot \left(\left(\left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right) \cdot \left(e^{-6.5 - z} \cdot {\left(z + 6.5\right)}^{\left(z + -0.5\right)}\right)\right) \]

Alternative 9
Error	3.8
Cost	29504

\[\sqrt{2 \cdot \pi} \cdot \left(\left(\left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{z}\right) + \left(\frac{-1259.1392167224028}{z + 1} + \frac{771.3234287776531}{z + 2}\right)\right)\right) \cdot \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\right) \]

Alternative 10
Error	48.9
Cost	29252

\[\begin{array}{l} t_0 := \frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\\ t_1 := \sqrt{2 \cdot \pi}\\ t_2 := \frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}}\\ t_3 := 0.9999999999998099 + \frac{188.7045801771354}{z}\\ \mathbf{if}\;z \leq 11.8:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(t_0 + t_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(t_0 + \left(t_3 + \frac{-283.5076408329034}{z \cdot z}\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	50.0
Cost	28736

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right) + \left(0.9999999999998099 + \frac{188.7045801771354}{z}\right)\right)\right) \]

Alternative 12
Error	53.4
Cost	28608

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{{\left(z + 6.5\right)}^{\left(z + -0.5\right)}}{e^{z + 6.5}} \cdot \left(\frac{676.5203681218851}{z} + \left(\frac{-176.6150291621406}{z + 3} + \left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right)\right)\right)\right) \]

Alternative 13
Error	56.3
Cost	28420

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ t_1 := \frac{\sqrt{0.15384615384615385}}{e^{6.5}}\\ t_2 := \left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\\ \mathbf{if}\;z \leq 3:\\ \;\;\;\;t_0 \cdot \left(\left(0.9999999999998099 + \left(t_2 + \left(\frac{529.8450874864218}{z \cdot z} + \frac{-176.6150291621406}{z}\right)\right)\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(0.9999999999998099 + \left(t_2 + \left(z \cdot 19.623892129126734 + -58.8716763873802\right)\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	59.5
Cost	28032

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \left(z \cdot 19.623892129126734 + -58.8716763873802\right)\right)\right)\right) \]

Alternative 15
Error	63.1
Cost	27904

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(\left(\left(\frac{-0.13857109526572012}{z + 5} + \left(\frac{12.507343278686905}{z + 4} + \frac{9.984369578019572 \cdot 10^{-6}}{z + 6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{z + 7}\right) + \frac{-176.6150291621406}{z}\right)\right)\right) \]

Alternative 16
Error	63.1
Cost	26880

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(-58.8716763873802 + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + \frac{12.368782167790762}{z}\right)\right)\right)\right) \]

Alternative 17
Error	63.1
Cost	26752

\[\sqrt{2 \cdot \pi} \cdot \left(\frac{\sqrt{0.15384615384615385}}{e^{6.5}} \cdot \left(0.9999999999998099 + \left(-58.8716763873802 + \left(\frac{1.5056327351493116 \cdot 10^{-7}}{z + 7} + 3.0991232646801787\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022295 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  :precision binary64
  :pre (> z 0.5)
  (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- z 1.0) 7.0) 0.5) (+ (- z 1.0) 0.5))) (exp (- (+ (+ (- z 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1.0) 1.0))) (/ -1259.1392167224028 (+ (- z 1.0) 2.0))) (/ 771.3234287776531 (+ (- z 1.0) 3.0))) (/ -176.6150291621406 (+ (- z 1.0) 4.0))) (/ 12.507343278686905 (+ (- z 1.0) 5.0))) (/ -0.13857109526572012 (+ (- z 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- z 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- z 1.0) 8.0)))))