Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 97.3%

Time: 1.7min

Alternatives: 7

Speedup: 1.5×

• could not determine a ground truth

  1. z = -4.858939463670953e+292

Specification

\[z \leq 0.5\]

Math

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

FPCore

(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 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 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) {
	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, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static 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 (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (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)
	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(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + 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

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

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[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $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]]]]

Initial Program: 96.6% accurate, 1.0× speedup

Math

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

FPCore

(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 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 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) {
	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, (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Java

public static 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 (Math.PI / Math.sin((Math.PI * z))) * (((Math.sqrt((Math.PI * 2.0)) * Math.pow(t_2, (t_0 + 0.5))) * Math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
}

Python

def code(z):
	t_0 = (1.0 - z) - 1.0
	t_1 = t_0 + 7.0
	t_2 = t_1 + 0.5
	return (math.pi / math.sin((math.pi * z))) * (((math.sqrt((math.pi * 2.0)) * math.pow(t_2, (t_0 + 0.5))) * math.exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (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)
	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(t_0 + 0.5))) * exp(Float64(-t_2))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(t_0 + 1.0))) + Float64(-1259.1392167224028 / Float64(t_0 + 2.0))) + Float64(771.3234287776531 / Float64(t_0 + 3.0))) + Float64(-176.6150291621406 / Float64(t_0 + 4.0))) + 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

MATLAB

function tmp = code(z)
	t_0 = (1.0 - z) - 1.0;
	t_1 = t_0 + 7.0;
	t_2 = t_1 + 0.5;
	tmp = (pi / sin((pi * z))) * (((sqrt((pi * 2.0)) * (t_2 ^ (t_0 + 0.5))) * exp(-t_2)) * ((((((((0.9999999999998099 + (676.5203681218851 / (t_0 + 1.0))) + (-1259.1392167224028 / (t_0 + 2.0))) + (771.3234287776531 / (t_0 + 3.0))) + (-176.6150291621406 / (t_0 + 4.0))) + (12.507343278686905 / (t_0 + 5.0))) + (-0.13857109526572012 / (t_0 + 6.0))) + (9.984369578019572e-6 / t_1)) + (1.5056327351493116e-7 / (t_0 + 8.0))));
end

Wolfram

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[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(t$95$0 + 4.0), $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]]]]

Alternative 1: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\frac{-1259.1392167224028}{2 - z} + \left(890.4744204239008 + z \cdot 751.1845319967683\right)\right) + \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right)\right) \cdot \left(e^{-7.5} + \frac{e^{-7.5}}{z}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
  (*
   (+
    (+
     (/ -1259.1392167224028 (- 2.0 z))
     (+ 890.4744204239008 (* z 751.1845319967683)))
    (+
     (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
     (+ 2.4783734731930944 (* z 0.49644453405676175))))
   (+ (exp -7.5) (/ (exp -7.5) z)))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * ((((-1259.1392167224028 / (2.0 - z)) + (890.4744204239008 + (z * 751.1845319967683))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))) * (exp(-7.5) + (exp(-7.5) / z)));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * ((((-1259.1392167224028 / (2.0 - z)) + (890.4744204239008 + (z * 751.1845319967683))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))) * (Math.exp(-7.5) + (Math.exp(-7.5) / z)));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * ((((-1259.1392167224028 / (2.0 - z)) + (890.4744204239008 + (z * 751.1845319967683))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))) * (math.exp(-7.5) + (math.exp(-7.5) / z)))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(Float64(Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(890.4744204239008 + Float64(z * 751.1845319967683))) + Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(2.4783734731930944 + Float64(z * 0.49644453405676175)))) * Float64(exp(-7.5) + Float64(exp(-7.5) / z))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * ((((-1259.1392167224028 / (2.0 - z)) + (890.4744204239008 + (z * 751.1845319967683))) + (((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175)))) * (exp(-7.5) + (exp(-7.5) / z)));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(890.4744204239008 + N[(z * 751.1845319967683), $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[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 96.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(e^{-7.5} + \frac{e^{-7.5}}{z}\right) \cdot \left(\left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7 - z} + \frac{1.5056327351493116 \cdot 10^{-7}}{8 - z}\right) + \left(2.4783734731930944 + z \cdot 0.49644453405676175\right)\right) + \left(\frac{-1259.1392167224028}{2 - z} + 890.4744204239008\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
  (*
   (+ (exp -7.5) (/ (exp -7.5) z))
   (+
    (+
     (+ (/ 9.984369578019572e-6 (- 7.0 z)) (/ 1.5056327351493116e-7 (- 8.0 z)))
     (+ 2.4783734731930944 (* z 0.49644453405676175)))
    (+ (/ -1259.1392167224028 (- 2.0 z)) 890.4744204239008)))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * ((exp(-7.5) + (exp(-7.5) / z)) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((-1259.1392167224028 / (2.0 - z)) + 890.4744204239008)));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * ((Math.exp(-7.5) + (Math.exp(-7.5) / z)) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((-1259.1392167224028 / (2.0 - z)) + 890.4744204239008)));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * ((math.exp(-7.5) + (math.exp(-7.5) / z)) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((-1259.1392167224028 / (2.0 - z)) + 890.4744204239008)))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(Float64(exp(-7.5) + Float64(exp(-7.5) / z)) * Float64(Float64(Float64(Float64(9.984369578019572e-6 / Float64(7.0 - z)) + Float64(1.5056327351493116e-7 / Float64(8.0 - z))) + Float64(2.4783734731930944 + Float64(z * 0.49644453405676175))) + Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + 890.4744204239008))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * ((exp(-7.5) + (exp(-7.5) / z)) * ((((9.984369578019572e-6 / (7.0 - z)) + (1.5056327351493116e-7 / (8.0 - z))) + (2.4783734731930944 + (z * 0.49644453405676175))) + ((-1259.1392167224028 / (2.0 - z)) + 890.4744204239008)));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(9.984369578019572e-6 / N[(7.0 - z), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(8.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.4783734731930944 + N[(z * 0.49644453405676175), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + 890.4744204239008), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 3: 95.9% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  263.3831869810514
  (* (sqrt PI) (/ (* (exp -7.5) (* (sqrt 2.0) (sqrt 7.5))) z))))

C

double code(double z) {
	return 263.3831869810514 * (sqrt(((double) M_PI)) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / z));
}

Java

public static double code(double z) {
	return 263.3831869810514 * (Math.sqrt(Math.PI) * ((Math.exp(-7.5) * (Math.sqrt(2.0) * Math.sqrt(7.5))) / z));
}

Python

def code(z):
	return 263.3831869810514 * (math.sqrt(math.pi) * ((math.exp(-7.5) * (math.sqrt(2.0) * math.sqrt(7.5))) / z))

Julia

function code(z)
	return Float64(263.3831869810514 * Float64(sqrt(pi) * Float64(Float64(exp(-7.5) * Float64(sqrt(2.0) * sqrt(7.5))) / z)))
end

MATLAB

function tmp = code(z)
	tmp = 263.3831869810514 * (sqrt(pi) * ((exp(-7.5) * (sqrt(2.0) * sqrt(7.5))) / z));
end

Wolfram

code[z_] := N[(263.3831869810514 * N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 4: 96.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (/ (exp -7.5) z) (* (sqrt 2.0) (sqrt 7.5)))
  (* (sqrt PI) 263.3831869810514)))

C

double code(double z) {
	return ((exp(-7.5) / z) * (sqrt(2.0) * sqrt(7.5))) * (sqrt(((double) M_PI)) * 263.3831869810514);
}

Java

public static double code(double z) {
	return ((Math.exp(-7.5) / z) * (Math.sqrt(2.0) * Math.sqrt(7.5))) * (Math.sqrt(Math.PI) * 263.3831869810514);
}

Python

def code(z):
	return ((math.exp(-7.5) / z) * (math.sqrt(2.0) * math.sqrt(7.5))) * (math.sqrt(math.pi) * 263.3831869810514)

Julia

function code(z)
	return Float64(Float64(Float64(exp(-7.5) / z) * Float64(sqrt(2.0) * sqrt(7.5))) * Float64(sqrt(pi) * 263.3831869810514))
end

MATLAB

function tmp = code(z)
	tmp = ((exp(-7.5) / z) * (sqrt(2.0) * sqrt(7.5))) * (sqrt(pi) * 263.3831869810514);
end

Wolfram

code[z_] := N[(N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * 263.3831869810514), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 5: 95.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\frac{e^{-7.5}}{z} \cdot \left(\frac{-1259.1392167224028}{2 - z} + \left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{-176.6150291621406}{4 - z}\right)\right) + \left(\frac{771.3234287776531}{3 - z} + 2.4783749183520145\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
  (*
   (/ (exp -7.5) z)
   (+
    (/ -1259.1392167224028 (- 2.0 z))
    (+
     (+
      (/ 676.5203681218851 (- 1.0 z))
      (+ 0.9999999999998099 (/ -176.6150291621406 (- 4.0 z))))
     (+ (/ 771.3234287776531 (- 3.0 z)) 2.4783749183520145))))))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * ((exp(-7.5) / z) * ((-1259.1392167224028 / (2.0 - z)) + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-176.6150291621406 / (4.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + 2.4783749183520145))));
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * ((Math.exp(-7.5) / z) * ((-1259.1392167224028 / (2.0 - z)) + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-176.6150291621406 / (4.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + 2.4783749183520145))));
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * ((math.exp(-7.5) / z) * ((-1259.1392167224028 / (2.0 - z)) + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-176.6150291621406 / (4.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + 2.4783749183520145))))

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(Float64(exp(-7.5) / z) * Float64(Float64(-1259.1392167224028 / Float64(2.0 - z)) + Float64(Float64(Float64(676.5203681218851 / Float64(1.0 - z)) + Float64(0.9999999999998099 + Float64(-176.6150291621406 / Float64(4.0 - z)))) + Float64(Float64(771.3234287776531 / Float64(3.0 - z)) + 2.4783749183520145)))))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * ((exp(-7.5) / z) * ((-1259.1392167224028 / (2.0 - z)) + (((676.5203681218851 / (1.0 - z)) + (0.9999999999998099 + (-176.6150291621406 / (4.0 - z)))) + ((771.3234287776531 / (3.0 - z)) + 2.4783749183520145))));
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * N[(N[(-1259.1392167224028 / N[(2.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(0.9999999999998099 + N[(-176.6150291621406 / N[(4.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(771.3234287776531 / N[(3.0 - z), $MachinePrecision]), $MachinePrecision] + 2.4783749183520145), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 6: 14.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{-7.5} \cdot \frac{\sqrt{\pi \cdot 2}}{\frac{z}{3.4783749183518244 \cdot {\left(z + 7.5\right)}^{\left(z + 0.5\right)}}} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (exp -7.5)
  (/
   (sqrt (* PI 2.0))
   (/ z (* 3.4783749183518244 (pow (+ z 7.5) (+ z 0.5)))))))

C

double code(double z) {
	return exp(-7.5) * (sqrt((((double) M_PI) * 2.0)) / (z / (3.4783749183518244 * pow((z + 7.5), (z + 0.5)))));
}

Java

public static double code(double z) {
	return Math.exp(-7.5) * (Math.sqrt((Math.PI * 2.0)) / (z / (3.4783749183518244 * Math.pow((z + 7.5), (z + 0.5)))));
}

Python

def code(z):
	return math.exp(-7.5) * (math.sqrt((math.pi * 2.0)) / (z / (3.4783749183518244 * math.pow((z + 7.5), (z + 0.5)))))

Julia

function code(z)
	return Float64(exp(-7.5) * Float64(sqrt(Float64(pi * 2.0)) / Float64(z / Float64(3.4783749183518244 * (Float64(z + 7.5) ^ Float64(z + 0.5))))))
end

MATLAB

function tmp = code(z)
	tmp = exp(-7.5) * (sqrt((pi * 2.0)) / (z / (3.4783749183518244 * ((z + 7.5) ^ (z + 0.5)))));
end

Wolfram

code[z_] := N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] / N[(z / N[(3.4783749183518244 * N[Power[N[(z + 7.5), $MachinePrecision], N[(z + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 14.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\pi \cdot 2} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\frac{e^{-7.5}}{z} \cdot 3.4783749183518244\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* (sqrt (* PI 2.0)) (pow (- 7.5 z) (- 0.5 z)))
  (* (/ (exp -7.5) z) 3.4783749183518244)))

C

double code(double z) {
	return (sqrt((((double) M_PI) * 2.0)) * pow((7.5 - z), (0.5 - z))) * ((exp(-7.5) / z) * 3.4783749183518244);
}

Java

public static double code(double z) {
	return (Math.sqrt((Math.PI * 2.0)) * Math.pow((7.5 - z), (0.5 - z))) * ((Math.exp(-7.5) / z) * 3.4783749183518244);
}

Python

def code(z):
	return (math.sqrt((math.pi * 2.0)) * math.pow((7.5 - z), (0.5 - z))) * ((math.exp(-7.5) / z) * 3.4783749183518244)

Julia

function code(z)
	return Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(7.5 - z) ^ Float64(0.5 - z))) * Float64(Float64(exp(-7.5) / z) * 3.4783749183518244))
end

MATLAB

function tmp = code(z)
	tmp = (sqrt((pi * 2.0)) * ((7.5 - z) ^ (0.5 - z))) * ((exp(-7.5) / z) * 3.4783749183518244);
end

Wolfram

code[z_] := N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(7.5 - z), $MachinePrecision], N[(0.5 - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[-7.5], $MachinePrecision] / z), $MachinePrecision] * 3.4783749183518244), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))