Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 1.4min

Precision: binary64

Cost: 70336

\[z \leq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-1259.1392167224028}{2 - z}\\ t_1 := 0.9999999999998099 + t_0\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {t_1}^{3}}{\mathsf{fma}\left(t_1, 0.9999999999998099 + \left(t_0 + \frac{-676.5203681218851}{1 - z}\right), \frac{\frac{457679.80848377093}{1 - z}}{1 - z}\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (z)
 :precision binary64
 (let* ((t_0 (/ -1259.1392167224028 (- 2.0 z)))
        (t_1 (+ 0.9999999999998099 t_0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (sqrt (* PI 2.0))
     (*
      (* (pow (- 7.5 z) (- 0.5 z)) (exp (+ z -7.5)))
      (+
       (/
        (+ (pow (/ 676.5203681218851 (- 1.0 z)) 3.0) (pow t_1 3.0))
        (fma
         t_1
         (+ 0.9999999999998099 (+ t_0 (/ -676.5203681218851 (- 1.0 z))))
         (/ (/ 457679.80848377093 (- 1.0 z)) (- 1.0 z))))
       (+
        (+
         (/ 9.984369578019572e-6 (- 7.0 z))
         (/ 1.5056327351493116e-7 (- 8.0 z)))
        (+
         (/ 771.3234287776531 (- 3.0 z))
         (+
          (/ -176.6150291621406 (- 4.0 z))
          (+
           (/ 12.507343278686905 (- 5.0 z))
           (/ -0.13857109526572012 (- 6.0 z))))))))))))

C

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

↓

double code(double z) {
	double t_0 = -1259.1392167224028 / (2.0 - z);
	double t_1 = 0.9999999999998099 + t_0;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (sqrt((((double) M_PI) * 2.0)) * ((pow((7.5 - z), (0.5 - z)) * exp((z + -7.5))) * (((pow((676.5203681218851 / (1.0 - z)), 3.0) + pow(t_1, 3.0)) / fma(t_1, (0.9999999999998099 + (t_0 + (-676.5203681218851 / (1.0 - z)))), ((457679.80848377093 / (1.0 - z)) / (1.0 - z)))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + ((771.3234287776531 / (3.0 - z)) + ((-176.6150291621406 / (4.0 - z)) + ((12.507343278686905 / (5.0 - z)) + (-0.13857109526572012 / (6.0 - z)))))))));
}

Julia

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

↓

function code(z)
	t_0 = Float64(-1259.1392167224028 / Float64(2.0 - z))
	t_1 = Float64(0.9999999999998099 + t_0)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(sqrt(Float64(pi * 2.0)) * Float64(Float64((Float64(7.5 - z) ^ Float64(0.5 - z)) * exp(Float64(z + -7.5))) * Float64(Float64(Float64((Float64(676.5203681218851 / Float64(1.0 - z)) ^ 3.0) + (t_1 ^ 3.0)) / fma(t_1, Float64(0.9999999999998099 + Float64(t_0 + Float64(-676.5203681218851 / Float64(1.0 - z)))), Float64(Float64(457679.80848377093 / Float64(1.0 - z)) / Float64(1.0 - z)))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + Float64(Float64(-176.6150291621406 / Float64(4.0 - z)) + Float64(Float64(12.507343278686905 / Float64(5.0 - z)) + Float64(-0.13857109526572012 / Float64(6.0 - z))))))))))
end

Wolfram

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

↓

code[z_] := Block[{t$95$0 = N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.9999999999998099 + t$95$0), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(z + -7.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(0.9999999999998099 + N[(t$95$0 + N[(-676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(457679.80848377093 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(12.507343278686905 / N[(5.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(6.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 1.7

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

2. Applied egg-rr 0.5

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)} \]

3. Simplified 1.0

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

   [Start] 0.5

   \[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \]

   distribute-lft-out [=>] 0.5

   \[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \color{blue}{\left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot e^{-\left(\left(\left(1 - z\right) + -1\right) + 7.5\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right)} \]

4. Applied egg-rr 0.5

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

5. Simplified 0.5

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\color{blue}{\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, 0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{1 - z}\right), \frac{\frac{457679.80848377093}{1 - z}}{1 - z}\right)}} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   Proof

   [Start] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z} + \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) - \frac{676.5203681218851}{1 - z} \cdot \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   +-commutative [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\color{blue}{\left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) - \frac{676.5203681218851}{1 - z} \cdot \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z}}} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   distribute-rgt-out-- [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\color{blue}{\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) \cdot \left(\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) - \frac{676.5203681218851}{1 - z}\right)} + \frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z}} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   fma-def [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, \left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right) - \frac{676.5203681218851}{1 - z}, \frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z}\right)}} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   associate--l+ [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, \color{blue}{0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{1 - z}\right)}, \frac{676.5203681218851}{1 - z} \cdot \frac{676.5203681218851}{1 - z}\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   associate-*l/ [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, 0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{1 - z}\right), \color{blue}{\frac{676.5203681218851 \cdot \frac{676.5203681218851}{1 - z}}{1 - z}}\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   associate-*r/ [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, 0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{1 - z}\right), \frac{\color{blue}{\frac{676.5203681218851 \cdot 676.5203681218851}{1 - z}}}{1 - z}\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]
   metadata-eval [=>] 0.5	\[ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, 0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} - \frac{676.5203681218851}{1 - z}\right), \frac{\frac{\color{blue}{457679.80848377093}}{1 - z}}{1 - z}\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]

6. Final simplification 0.5

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right) \cdot \left(\frac{{\left(\frac{676.5203681218851}{1 - z}\right)}^{3} + {\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}\right)}^{3}}{\mathsf{fma}\left(0.9999999999998099 + \frac{-1259.1392167224028}{2 - z}, 0.9999999999998099 + \left(\frac{-1259.1392167224028}{2 - z} + \frac{-676.5203681218851}{1 - z}\right), \frac{\frac{457679.80848377093}{1 - z}}{1 - z}\right)} + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(\frac{771.3234287776531}{3 - z} + \left(\frac{-176.6150291621406}{4 - z} + \left(\frac{12.507343278686905}{5 - z} + \frac{-0.13857109526572012}{6 - z}\right)\right)\right)\right)\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	50368

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

Alternative 2
Error	0.5
Cost	50112

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

Alternative 3
Error	1.0
Cost	49088

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

Alternative 4
Error	1.0
Cost	49088

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

Alternative 5
Error	1.1
Cost	47424

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(263.4062807184368 + z \cdot 436.9000215473151\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \frac{-0.13857109526572012}{6 - z}\right)\right)\right) \]

Alternative 6
Error	1.5
Cost	47296

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z + -7.5} \cdot \left(\left(0.9999999999998099 + \left(304.05856935323476 + z \cdot 447.4381671388014\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + -41.67538381734206\right)\right)\right)\right) \]

Alternative 7
Error	1.7
Cost	32640

\[263.3831869810514 \cdot \left(\left(\frac{\sqrt{2}}{z} \cdot \left(e^{-7.5} \cdot \sqrt{7.5}\right)\right) \cdot \sqrt{\pi}\right) \]

Alternative 8
Error	2.3
Cost	27200

\[\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(e^{z + -7.5} \cdot \left(\frac{676.5203681218851}{1 - z} + -413.1371811408337\right)\right)\right)\right) \]

Alternative 9
Error	55.0
Cost	26368

\[\frac{1}{z} \cdot \left(\sqrt{\pi \cdot 2} \cdot \frac{0.9999999999998099 \cdot \sqrt{7.5}}{e^{7.5}}\right) \]

Reproduce

herbie shell --seed 2023237 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))