Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.7 → 0.5

Time: 1.6min

Precision: binary64

Cost: 77376

\[z \leq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t_0 + 7\\ t_2 := t_1 + 0.5\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t_2}^{\left(\mathsf{fma}\left(\sqrt[3]{1 - z}, \sqrt[3]{{\left(1 - z\right)}^{2}}, -1\right) + 0.5\right)}\right) \cdot e^{-t_2}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{3 - \left(1 + z\right)}\right) + \left(\frac{771.3234287776531}{4 - \left(1 + z\right)} + \frac{-176.6150291621406}{5 - \left(1 + z\right)}\right)\right)\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t_0 + 8}\right)\right) \end{array} \]

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 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (*
       (sqrt (* PI 2.0))
       (pow
        t_2
        (+ (fma (cbrt (- 1.0 z)) (cbrt (pow (- 1.0 z) 2.0)) -1.0) 0.5)))
      (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          0.9999999999998099
          (+
           (+
            (/ 676.5203681218851 (- 1.0 z))
            (/ -1259.1392167224028 (- 3.0 (+ 1.0 z))))
           (+
            (/ 771.3234287776531 (- 4.0 (+ 1.0 z)))
            (/ -176.6150291621406 (- 5.0 (+ 1.0 z))))))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

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 = (1.0 - z) - 1.0;
	double t_1 = t_0 + 7.0;
	double t_2 = t_1 + 0.5;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (fma(cbrt((1.0 - z)), cbrt(pow((1.0 - z), 2.0)), -1.0) + 0.5))) * exp(-t_2)) * (((((0.9999999999998099 + (((676.5203681218851 / (1.0 - z)) + (-1259.1392167224028 / (3.0 - (1.0 + z)))) + ((771.3234287776531 / (4.0 - (1.0 + z))) + (-176.6150291621406 / (5.0 - (1.0 + z)))))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

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(Float64(1.0 - z) - 1.0)
	t_1 = Float64(t_0 + 7.0)
	t_2 = Float64(t_1 + 0.5)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(fma(cbrt(Float64(1.0 - z)), cbrt((Float64(1.0 - z) ^ 2.0)), -1.0) + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(-1259.1392167224028 / Float64(3.0 - Float64(1.0 + z)))) + Float64(Float64(771.3234287776531 / Float64(4.0 - Float64(1.0 + z))) + Float64(-176.6150291621406 / Float64(5.0 - Float64(1.0 + z)))))) + Float64(12.507343278686905 / Float64(t_0 + 5.0))) + Float64(-0.13857109526572012 / Float64(t_0 + 6.0))) + Float64(9.984369578019572e-6 / t_1)) + Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0)))))
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[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 0.5), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(N[(N[Power[N[(1.0 - z), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[(1.0 - z), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.9999999999998099 + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(3.0 - N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(4.0 - N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(5.0 - N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(t$95$0 + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $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

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

3. Applied egg-rr 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	51584

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

Alternative 2
Error	0.5
Cost	50112

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

Alternative 3
Error	0.8
Cost	49920

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

Alternative 4
Error	0.8
Cost	49920

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

Alternative 5
Error	1.4
Cost	49664

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

Alternative 6
Error	1.4
Cost	49664

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

Alternative 7
Error	2.1
Cost	49536

\[\begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot e^{z + -7.5}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(46.9507597606837 + \left(\frac{771.3234287776531}{4 - \left(1 + z\right)} + \frac{-176.6150291621406}{5 - \left(1 + z\right)}\right)\right)\right) + \frac{12.507343278686905}{t_0 + 5}\right) + \frac{-0.13857109526572012}{t_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \end{array} \]

Alternative 8
Error	2.2
Cost	49024

\[\frac{\pi \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot e^{\left(z + -7.5\right) + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{2 - z}\right) + 212.9540523020159\right)\right) + \frac{12.507343278686905}{1 - \left(z + -4\right)}\right) + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{1 - \left(z + -7\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 9
Error	2.2
Cost	49024

\[\frac{\pi \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot e^{\left(\left(0.5 - z\right) \cdot \log \left(7.5 - z\right) + z\right) - 7.5}\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-676.5203681218851}{z + -1} + \frac{-1259.1392167224028}{2 - z}\right) + 212.9540523020159\right)\right) + \frac{12.507343278686905}{1 - \left(z + -4\right)}\right) + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{1 - \left(z + -7\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 10
Error	2.4
Cost	48128

\[\frac{\pi \cdot \left(\left(\sqrt{2 \cdot \pi} \cdot e^{\left(z + -7.5\right) + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(260.9048120626994 + \frac{12.507343278686905}{1 - \left(z + -4\right)}\right) + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{1 - \left(z + -7\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 11
Error	2.6
Cost	48064

\[\frac{\pi \cdot \left(\left(\left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \pi}\right) \cdot e^{z + -7.5}\right) \cdot \left(\left(\left(\left(260.9048120626994 + \frac{12.507343278686905}{1 - \left(z + -4\right)}\right) + \frac{-0.13857109526572012}{1 - \left(z + -5\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{1 - \left(z + -6\right)}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{1 - \left(z + -7\right)}\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 12
Error	2.6
Cost	48064

\[\frac{\pi \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{z + -7.5} \cdot \left(\left(\left(\left(260.9048120626994 + \frac{-12.507343278686905}{\left(z + -4\right) - 1}\right) + \frac{0.13857109526572012}{\left(z + -5\right) - 1}\right) + \frac{-9.984369578019572 \cdot 10^{-6}}{\left(z + -6\right) - 1}\right) + \frac{-1.5056327351493116 \cdot 10^{-7}}{\left(z + -7\right) - 1}\right)\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 13
Error	45.8
Cost	46080

\[\frac{260.9048120626994 \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{\left(z - 7.5\right) + \log \left(7.5 - z\right) \cdot \left(0.5 - z\right)} \cdot \pi\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 14
Error	45.8
Cost	46016

\[\frac{260.9048120626994 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{z - 7.5} \cdot \pi\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 15
Error	45.8
Cost	46016

\[\frac{\pi \cdot \left(260.9048120626994 \cdot \left({\left(7.5 - z\right)}^{\left(0.5 - z\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot e^{z - 7.5}\right)\right)\right)}{\sin \left(\pi \cdot z\right)} \]

Alternative 16
Error	52.8
Cost	32704

\[260.9048120626994 \cdot \left(\sqrt{2 \cdot \pi} \cdot \left(e^{-7.5} \cdot \frac{\sqrt{7.5} \cdot \pi}{z}\right)\right) \]

Reproduce

herbie shell --seed 2023010 
(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))))))