Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 99.2%

Time: 19.3s

Alternatives: 8

Speedup: N/A×

• could not determine a ground truth

  1. z = -91870436080481.56

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: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ t_3 := \frac{\pi}{\sin \left(\pi \cdot z\right)}\\ t_4 := \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\\ t_5 := \sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\\ t_6 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_7 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_8 := \frac{12.507343278686905}{\left(1 - z\right) - -4}\\ t_9 := t\_5 \cdot e^{-1 \cdot t\_2}\\ t_10 := \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\\ \mathbf{if}\;t\_3 \cdot \left(t\_9 \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + t\_7\right) + t\_10\right) + t\_6\right)\right) \leq 10^{+285}:\\ \;\;\;\;t\_3 \cdot \left(t\_9 \cdot \left(\left(\frac{{t\_4}^{3} + {t\_8}^{3}}{\mathsf{fma}\left(t\_4, t\_4, t\_8 \cdot t\_8 - t\_4 \cdot t\_8\right)} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + t\_6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(\left(t\_5 \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right) + t\_7\right) + t\_10\right) + t\_6\right)\right)\\ \end{array} \end{array} \]

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))
        (t_3 (/ PI (sin (* PI z))))
        (t_4
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
            (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
           (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
          (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
        (t_5 (* (sqrt (* PI 2.0)) (pow t_2 (+ t_0 0.5))))
        (t_6 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_7 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_8 (/ 12.507343278686905 (- (- 1.0 z) -4.0)))
        (t_9 (* t_5 (exp (* -1.0 t_2))))
        (t_10 (/ 9.984369578019572e-6 t_1)))
   (if (<=
        (*
         t_3
         (*
          t_9
          (+
           (+
            (+
             (+
              (+
               (+
                (+
                 (+ 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)))
             t_7)
            t_10)
           t_6)))
        1e+285)
     (*
      t_3
      (*
       t_9
       (+
        (+
         (/
          (+ (pow t_4 3.0) (pow t_8 3.0))
          (fma t_4 t_4 (- (* t_8 t_8) (* t_4 t_8))))
         (+
          (/ -0.13857109526572012 (- (- 1.0 z) -5.0))
          (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))))
        t_6)))
     (*
      t_3
      (*
       (* t_5 (+ (exp -7.5) (fma (exp -7.5) z (* (* 0.5 (* z (exp -7.5))) z))))
       (+
        (+
         (+
          (+
           263.4062807184368
           (* z (+ 436.9000215473151 (* 545.0359493463282 z))))
          t_7)
         t_10)
        t_6))))))

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;
	double t_3 = ((double) M_PI) / sin((((double) M_PI) * z));
	double t_4 = (((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (771.3234287776531 / ((1.0 - z) - -2.0))) + (-176.6150291621406 / ((1.0 - z) - -3.0));
	double t_5 = sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_0 + 0.5));
	double t_6 = 1.5056327351493116e-7 / (t_0 + 8.0);
	double t_7 = -0.13857109526572012 / (t_0 + 6.0);
	double t_8 = 12.507343278686905 / ((1.0 - z) - -4.0);
	double t_9 = t_5 * exp((-1.0 * t_2));
	double t_10 = 9.984369578019572e-6 / t_1;
	double tmp;
	if ((t_3 * (t_9 * ((((((((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))) + t_7) + t_10) + t_6))) <= 1e+285) {
		tmp = t_3 * (t_9 * ((((pow(t_4, 3.0) + pow(t_8, 3.0)) / fma(t_4, t_4, ((t_8 * t_8) - (t_4 * t_8)))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + t_6));
	} else {
		tmp = t_3 * ((t_5 * (exp(-7.5) + fma(exp(-7.5), z, ((0.5 * (z * exp(-7.5))) * z)))) * ((((263.4062807184368 + (z * (436.9000215473151 + (545.0359493463282 * z)))) + t_7) + t_10) + t_6));
	}
	return tmp;
}

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)
	t_3 = Float64(pi / sin(Float64(pi * z)))
	t_4 = Float64(Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0))) + Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0)))
	t_5 = Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_0 + 0.5)))
	t_6 = Float64(1.5056327351493116e-7 / Float64(t_0 + 8.0))
	t_7 = Float64(-0.13857109526572012 / Float64(t_0 + 6.0))
	t_8 = Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))
	t_9 = Float64(t_5 * exp(Float64(-1.0 * t_2)))
	t_10 = Float64(9.984369578019572e-6 / t_1)
	tmp = 0.0
	if (Float64(t_3 * Float64(t_9 * 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))) + t_7) + t_10) + t_6))) <= 1e+285)
		tmp = Float64(t_3 * Float64(t_9 * Float64(Float64(Float64(Float64((t_4 ^ 3.0) + (t_8 ^ 3.0)) / fma(t_4, t_4, Float64(Float64(t_8 * t_8) - Float64(t_4 * t_8)))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + t_6)));
	else
		tmp = Float64(t_3 * Float64(Float64(t_5 * Float64(exp(-7.5) + fma(exp(-7.5), z, Float64(Float64(0.5 * Float64(z * exp(-7.5))) * z)))) * Float64(Float64(Float64(Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(545.0359493463282 * z)))) + t_7) + t_10) + t_6)));
	end
	return tmp
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]}, Block[{t$95$3 = N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(1.5056327351493116e-7 / N[(t$95$0 + 8.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(-0.13857109526572012 / N[(t$95$0 + 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$5 * N[Exp[N[(-1.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(9.984369578019572e-6 / t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[(t$95$9 * 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] + t$95$7), $MachinePrecision] + t$95$10), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+285], N[(t$95$3 * N[(t$95$9 * N[(N[(N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] + N[Power[t$95$8, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + N[(N[(t$95$8 * t$95$8), $MachinePrecision] - N[(t$95$4 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[(t$95$5 * N[(N[Exp[-7.5], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * z + N[(N[(0.5 * N[(z * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(545.0359493463282 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$7), $MachinePrecision] + t$95$10), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64)))))) < 9.9999999999999998e284

   1. Initial program 97.3%

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

   4. Applied rewrites 98.4%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\color{blue}{\frac{{\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)}^{3} + {\left(\frac{12.507343278686905}{\left(1 - z\right) - -4}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}, \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}, \frac{12.507343278686905}{\left(1 - z\right) - -4} \cdot \frac{12.507343278686905}{\left(1 - z\right) - -4} - \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) \cdot \frac{12.507343278686905}{\left(1 - z\right) - -4}\right)}} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if 9.9999999999999998e284 < (*.f64 (/.f64 (PI.f64) (sin.f64 (*.f64 (PI.f64) z))) (*.f64 (*.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) #s(literal 2 binary64))) (pow.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64)) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1/2 binary64)))) (exp.f64 (neg.f64 (+.f64 (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)) #s(literal 1/2 binary64))))) (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 #s(literal 9999999999998099/10000000000000000 binary64) (/.f64 #s(literal 6765203681218851/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 1 binary64)))) (/.f64 #s(literal -3147848041806007/2500000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 2 binary64)))) (/.f64 #s(literal 7713234287776531/10000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 3 binary64)))) (/.f64 #s(literal -883075145810703/5000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 4 binary64)))) (/.f64 #s(literal 2501468655737381/200000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 5 binary64)))) (/.f64 #s(literal -3464277381643003/25000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 6 binary64)))) (/.f64 #s(literal 2496092394504893/250000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 7 binary64)))) (/.f64 #s(literal 3764081837873279/25000000000000000000000 binary64) (+.f64 (-.f64 (-.f64 #s(literal 1 binary64) z) #s(literal 1 binary64)) #s(literal 8 binary64))))))

   1. Initial program 39.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(e^{\frac{-15}{2}} + z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z} \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \left(e^{\frac{-15}{2}} \cdot z + \color{blue}{\left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, \color{blue}{z}, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      9. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + \color{blue}{z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \color{blue}{\left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \color{blue}{\frac{2943194126470171931171}{5400000000000000000} \cdot z}\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. lower-*.f64 99.7

         \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot \color{blue}{z}\right)\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right)} + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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^{-1 \cdot \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) \leq 10^{+285}:\\ \;\;\;\;\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^{-1 \cdot \left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\frac{{\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)}^{3} + {\left(\frac{12.507343278686905}{\left(1 - z\right) - -4}\right)}^{3}}{\mathsf{fma}\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}, \left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}, \frac{12.507343278686905}{\left(1 - z\right) - -4} \cdot \frac{12.507343278686905}{\left(1 - z\right) - -4} - \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) \cdot \frac{12.507343278686905}{\left(1 - z\right) - -4}\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(\left(1 - z\right) - 1\right) + 8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\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 \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\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)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\pi}^{0.5} \cdot {2}^{0.5}\right) \cdot {\left(t\_1 + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (* (pow PI 0.5) (pow 2.0 0.5)) (pow (+ t_1 0.5) (+ t_0 0.5)))
      (+ (exp -7.5) (fma (exp -7.5) z (* (* 0.5 (* z (exp -7.5))) z))))
     (+
      (+
       (+
        (+
         263.4062807184368
         (* z (+ 436.9000215473151 (* 545.0359493463282 z))))
        (/ -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;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * ((((pow(((double) M_PI), 0.5) * pow(2.0, 0.5)) * pow((t_1 + 0.5), (t_0 + 0.5))) * (exp(-7.5) + fma(exp(-7.5), z, ((0.5 * (z * exp(-7.5))) * z)))) * ((((263.4062807184368 + (z * (436.9000215473151 + (545.0359493463282 * z)))) + (-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)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(Float64((pi ^ 0.5) * (2.0 ^ 0.5)) * (Float64(t_1 + 0.5) ^ Float64(t_0 + 0.5))) * Float64(exp(-7.5) + fma(exp(-7.5), z, Float64(Float64(0.5 * Float64(z * exp(-7.5))) * z)))) * Float64(Float64(Float64(Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(545.0359493463282 * z)))) + 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_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $MachinePrecision]}, N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[Power[Pi, 0.5], $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * z + N[(N[(0.5 * N[(z * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(545.0359493463282 * z), $MachinePrecision]), $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 95.0%

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

3. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(e^{\frac{-15}{2}} + z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z} \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \left(e^{\frac{-15}{2}} \cdot z + \color{blue}{\left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, \color{blue}{z}, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + \color{blue}{z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \color{blue}{\left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \color{blue}{\frac{2943194126470171931171}{5400000000000000000} \cdot z}\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 97.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot \color{blue}{z}\right)\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 97.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right)} + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\pi \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)} \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot 2}} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. sqrt-prod N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. pow1/2 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\color{blue}{\pi}}^{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\pi}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lift-pow.f64 98.4

       \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\left({\pi}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 98.4%

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left({\pi}^{0.5} \cdot {2}^{0.5}\right)} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
11. Add Preprocessing

Alternative 3: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(t\_1 + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)))
   (*
    (/ PI (sin (* PI z)))
    (*
     (*
      (* (sqrt (* PI 2.0)) (pow (+ t_1 0.5) (+ t_0 0.5)))
      (+ (exp -7.5) (fma (exp -7.5) z (* (* 0.5 (* z (exp -7.5))) z))))
     (+
      (+
       (+
        (+
         263.4062807184368
         (* z (+ 436.9000215473151 (* 545.0359493463282 z))))
        (/ -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;
	return (((double) M_PI) / sin((((double) M_PI) * z))) * (((sqrt((((double) M_PI) * 2.0)) * pow((t_1 + 0.5), (t_0 + 0.5))) * (exp(-7.5) + fma(exp(-7.5), z, ((0.5 * (z * exp(-7.5))) * z)))) * ((((263.4062807184368 + (z * (436.9000215473151 + (545.0359493463282 * z)))) + (-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)
	return Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (Float64(t_1 + 0.5) ^ Float64(t_0 + 0.5))) * Float64(exp(-7.5) + fma(exp(-7.5), z, Float64(Float64(0.5 * Float64(z * exp(-7.5))) * z)))) * Float64(Float64(Float64(Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(545.0359493463282 * z)))) + 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_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 7.0), $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[N[(t$95$1 + 0.5), $MachinePrecision], N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[-7.5], $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * z + N[(N[(0.5 * N[(z * N[Exp[-7.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(545.0359493463282 * z), $MachinePrecision]), $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 95.0%

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

3. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(e^{\frac{-15}{2}} + z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z} \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \left(e^{\frac{-15}{2}} \cdot z + \color{blue}{\left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, \color{blue}{z}, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)\right)} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + \color{blue}{z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \color{blue}{\left(\frac{39321001939258358983}{90000000000000000} + \frac{2943194126470171931171}{5400000000000000000} \cdot z\right)}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \color{blue}{\frac{2943194126470171931171}{5400000000000000000} \cdot z}\right)\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 97.5

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot \color{blue}{z}\right)\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 97.5%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot \left(e^{-7.5} + \mathsf{fma}\left(e^{-7.5}, z, \left(0.5 \cdot \left(z \cdot e^{-7.5}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + 545.0359493463282 \cdot z\right)\right)} + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Add Preprocessing

Alternative 4: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := t\_0 + 0.5\\ t_2 := \left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\\ t_3 := \frac{-176.6150291621406}{\left(1 - z\right) - -3}\\ \left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left({\left(\pi \cdot 2\right)}^{0.5} \cdot {t\_1}^{\left(\left(1 - z\right) - 0.5\right)}\right) \cdot \frac{1}{e^{t\_1}}\right)\right) \cdot \left(\left(\left(\left(\frac{{t\_2}^{3} + {t\_3}^{3}}{\mathsf{fma}\left(t\_2, t\_2, t\_3 \cdot t\_3 - t\_2 \cdot t\_3\right)} + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0))
        (t_1 (+ t_0 0.5))
        (t_2
         (+
          (+
           (+ 0.9999999999998099 (/ 676.5203681218851 (- 1.0 z)))
           (/ -1259.1392167224028 (- (- 1.0 z) -1.0)))
          (/ 771.3234287776531 (- (- 1.0 z) -2.0))))
        (t_3 (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
   (*
    (*
     (/ PI (sin (* PI z)))
     (*
      (* (pow (* PI 2.0) 0.5) (pow t_1 (- (- 1.0 z) 0.5)))
      (/ 1.0 (exp t_1))))
    (+
     (+
      (+
       (+
        (/
         (+ (pow t_2 3.0) (pow t_3 3.0))
         (fma t_2 t_2 (- (* t_3 t_3) (* t_2 t_3))))
        (/ 12.507343278686905 (- (- 1.0 z) -4.0)))
       (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
      (/ 9.984369578019572e-6 t_0))
     (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))))

C

double code(double z) {
	double t_0 = (1.0 - z) - -6.0;
	double t_1 = t_0 + 0.5;
	double t_2 = ((0.9999999999998099 + (676.5203681218851 / (1.0 - z))) + (-1259.1392167224028 / ((1.0 - z) - -1.0))) + (771.3234287776531 / ((1.0 - z) - -2.0));
	double t_3 = -176.6150291621406 / ((1.0 - z) - -3.0);
	return ((((double) M_PI) / sin((((double) M_PI) * z))) * ((pow((((double) M_PI) * 2.0), 0.5) * pow(t_1, ((1.0 - z) - 0.5))) * (1.0 / exp(t_1)))) * ((((((pow(t_2, 3.0) + pow(t_3, 3.0)) / fma(t_2, t_2, ((t_3 * t_3) - (t_2 * t_3)))) + (12.507343278686905 / ((1.0 - z) - -4.0))) + (-0.13857109526572012 / ((1.0 - z) - -5.0))) + (9.984369578019572e-6 / t_0)) + (1.5056327351493116e-7 / ((1.0 - z) - -7.0)));
}

Julia

function code(z)
	t_0 = Float64(Float64(1.0 - z) - -6.0)
	t_1 = Float64(t_0 + 0.5)
	t_2 = Float64(Float64(Float64(0.9999999999998099 + Float64(676.5203681218851 / Float64(1.0 - z))) + Float64(-1259.1392167224028 / Float64(Float64(1.0 - z) - -1.0))) + Float64(771.3234287776531 / Float64(Float64(1.0 - z) - -2.0)))
	t_3 = Float64(-176.6150291621406 / Float64(Float64(1.0 - z) - -3.0))
	return Float64(Float64(Float64(pi / sin(Float64(pi * z))) * Float64(Float64((Float64(pi * 2.0) ^ 0.5) * (t_1 ^ Float64(Float64(1.0 - z) - 0.5))) * Float64(1.0 / exp(t_1)))) * Float64(Float64(Float64(Float64(Float64(Float64((t_2 ^ 3.0) + (t_3 ^ 3.0)) / fma(t_2, t_2, Float64(Float64(t_3 * t_3) - Float64(t_2 * t_3)))) + Float64(12.507343278686905 / Float64(Float64(1.0 - z) - -4.0))) + Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0))) + Float64(9.984369578019572e-6 / t_0)) + Float64(1.5056327351493116e-7 / Float64(Float64(1.0 - z) - -7.0))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.9999999999998099 + N[(676.5203681218851 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1259.1392167224028 / N[(N[(1.0 - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(771.3234287776531 / N[(N[(1.0 - z), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-176.6150291621406 / N[(N[(1.0 - z), $MachinePrecision] - -3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(Pi / N[Sin[N[(Pi * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(Pi * 2.0), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[t$95$1, N[(N[(1.0 - z), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(12.507343278686905 / N[(N[(1.0 - z), $MachinePrecision] - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(N[(1.0 - z), $MachinePrecision] - -7.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 95.0%

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

4. Applied rewrites 95.2%

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

6. Applied rewrites 96.4%

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

7. Final simplification 96.4%

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

Alternative 5: 97.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\\ t_1 := \left(1 - z\right) - 1\\ t_2 := \left(t\_1 + 7\right) + 0.5\\ \frac{1 + \left(z \cdot z\right) \cdot \left(\left(\left(-1 \cdot z\right) \cdot z\right) \cdot \left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, {\pi}^{6}, \mathsf{fma}\left(0.001388888888888889, {\pi}^{6}, 0.16666666666666666 \cdot \left(\left(\pi \cdot \pi\right) \cdot t\_0\right)\right)\right) + t\_0\right) - -0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {t\_2}^{\left(t\_1 + 0.5\right)}\right) \cdot e^{-1 \cdot t\_2}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_1 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0
         (fma
          -0.027777777777777776
          (pow PI 4.0)
          (* 0.008333333333333333 (pow PI 4.0))))
        (t_1 (- (- 1.0 z) 1.0))
        (t_2 (+ (+ t_1 7.0) 0.5)))
   (*
    (/
     (+
      1.0
      (*
       (* z z)
       (-
        (*
         (* (* -1.0 z) z)
         (+
          (*
           (* z z)
           (fma
            -0.0001984126984126984
            (pow PI 6.0)
            (fma
             0.001388888888888889
             (pow PI 6.0)
             (* 0.16666666666666666 (* (* PI PI) t_0)))))
          t_0))
        (* -0.16666666666666666 (* PI PI)))))
     z)
    (*
     (* (* (sqrt (* PI 2.0)) (pow t_2 (+ t_1 0.5))) (exp (* -1.0 t_2)))
     (+
      (+
       (+
        263.4062807184368
        (*
         z
         (+
          436.9000215473151
          (* z (+ 545.0359493463282 (* 606.6767878347069 z))))))
       (+
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))
        (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))))
      (/ 1.5056327351493116e-7 (+ t_1 8.0)))))))

C

double code(double z) {
	double t_0 = fma(-0.027777777777777776, pow(((double) M_PI), 4.0), (0.008333333333333333 * pow(((double) M_PI), 4.0)));
	double t_1 = (1.0 - z) - 1.0;
	double t_2 = (t_1 + 7.0) + 0.5;
	return ((1.0 + ((z * z) * ((((-1.0 * z) * z) * (((z * z) * fma(-0.0001984126984126984, pow(((double) M_PI), 6.0), fma(0.001388888888888889, pow(((double) M_PI), 6.0), (0.16666666666666666 * ((((double) M_PI) * ((double) M_PI)) * t_0))))) + t_0)) - (-0.16666666666666666 * (((double) M_PI) * ((double) M_PI)))))) / z) * (((sqrt((((double) M_PI) * 2.0)) * pow(t_2, (t_1 + 0.5))) * exp((-1.0 * t_2))) * (((263.4062807184368 + (z * (436.9000215473151 + (z * (545.0359493463282 + (606.6767878347069 * z)))))) + ((-0.13857109526572012 / ((1.0 - z) - -5.0)) + (9.984369578019572e-6 / ((1.0 - z) - -6.0)))) + (1.5056327351493116e-7 / (t_1 + 8.0))));
}

Julia

function code(z)
	t_0 = fma(-0.027777777777777776, (pi ^ 4.0), Float64(0.008333333333333333 * (pi ^ 4.0)))
	t_1 = Float64(Float64(1.0 - z) - 1.0)
	t_2 = Float64(Float64(t_1 + 7.0) + 0.5)
	return Float64(Float64(Float64(1.0 + Float64(Float64(z * z) * Float64(Float64(Float64(Float64(-1.0 * z) * z) * Float64(Float64(Float64(z * z) * fma(-0.0001984126984126984, (pi ^ 6.0), fma(0.001388888888888889, (pi ^ 6.0), Float64(0.16666666666666666 * Float64(Float64(pi * pi) * t_0))))) + t_0)) - Float64(-0.16666666666666666 * Float64(pi * pi))))) / z) * Float64(Float64(Float64(sqrt(Float64(pi * 2.0)) * (t_2 ^ Float64(t_1 + 0.5))) * exp(Float64(-1.0 * t_2))) * Float64(Float64(Float64(263.4062807184368 + Float64(z * Float64(436.9000215473151 + Float64(z * Float64(545.0359493463282 + Float64(606.6767878347069 * z)))))) + Float64(Float64(-0.13857109526572012 / Float64(Float64(1.0 - z) - -5.0)) + Float64(9.984369578019572e-6 / Float64(Float64(1.0 - z) - -6.0)))) + Float64(1.5056327351493116e-7 / Float64(t_1 + 8.0)))))
end

Wolfram

code[z_] := Block[{t$95$0 = N[(-0.027777777777777776 * N[Power[Pi, 4.0], $MachinePrecision] + N[(0.008333333333333333 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + 7.0), $MachinePrecision] + 0.5), $MachinePrecision]}, N[(N[(N[(1.0 + N[(N[(z * z), $MachinePrecision] * N[(N[(N[(N[(-1.0 * z), $MachinePrecision] * z), $MachinePrecision] * N[(N[(N[(z * z), $MachinePrecision] * N[(-0.0001984126984126984 * N[Power[Pi, 6.0], $MachinePrecision] + N[(0.001388888888888889 * N[Power[Pi, 6.0], $MachinePrecision] + N[(0.16666666666666666 * N[(N[(Pi * Pi), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(Pi * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, N[(t$95$1 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(-1.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(263.4062807184368 + N[(z * N[(436.9000215473151 + N[(z * N[(545.0359493463282 + N[(606.6767878347069 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.13857109526572012 / N[(N[(1.0 - z), $MachinePrecision] - -5.0), $MachinePrecision]), $MachinePrecision] + N[(9.984369578019572e-6 / N[(N[(1.0 - z), $MachinePrecision] - -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5056327351493116e-7 / N[(t$95$1 + 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 95.0%

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

4. Applied rewrites 96.1%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(1 - z\right) - 0}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right)} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + \color{blue}{z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \color{blue}{\left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)}\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + \color{blue}{z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)}\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \color{blue}{\left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)}\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \color{blue}{\frac{196563279258445065194677}{324000000000000000000} \cdot z}\right)\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 95.9

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot \color{blue}{z}\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 95.9%

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\color{blue}{\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right)} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + {z}^{2} \cdot \left({z}^{2} \cdot \left(-1 \cdot \left({z}^{2} \cdot \left(\frac{-1}{5040} \cdot {\mathsf{PI}\left(\right)}^{6} + \left(\frac{1}{720} \cdot {\mathsf{PI}\left(\right)}^{6} + \frac{1}{6} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{36} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) - \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\frac{7902188421553103227}{30000000000000000} + z \cdot \left(\frac{39321001939258358983}{90000000000000000} + z \cdot \left(\frac{2943194126470171931171}{5400000000000000000} + \frac{196563279258445065194677}{324000000000000000000} \cdot z\right)\right)\right) + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6}\right)\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 96.1%

    \[\leadsto \color{blue}{\frac{1 + \left(z \cdot z\right) \cdot \left(\left(z \cdot z\right) \cdot \left(-1 \cdot \left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, {\pi}^{6}, \mathsf{fma}\left(0.001388888888888889, {\pi}^{6}, 0.16666666666666666 \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\right)\right)\right)\right) - \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\right) - -0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)}{z}} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Final simplification 96.1%

    \[\leadsto \frac{1 + \left(z \cdot z\right) \cdot \left(\left(\left(-1 \cdot z\right) \cdot z\right) \cdot \left(\left(z \cdot z\right) \cdot \mathsf{fma}\left(-0.0001984126984126984, {\pi}^{6}, \mathsf{fma}\left(0.001388888888888889, {\pi}^{6}, 0.16666666666666666 \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\right)\right)\right) + \mathsf{fma}\left(-0.027777777777777776, {\pi}^{4}, 0.008333333333333333 \cdot {\pi}^{4}\right)\right) - -0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right)}{z} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-1 \cdot \left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(263.4062807184368 + z \cdot \left(436.9000215473151 + z \cdot \left(545.0359493463282 + 606.6767878347069 \cdot z\right)\right)\right) + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
12. Add Preprocessing

Alternative 6: 96.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \log 7.5 - 0.06666666666666667\\ t_1 := e^{-7.5} \cdot \sqrt{15}\\ t_2 := \frac{t\_1}{\pi}\\ t_3 := 436.8961725563396 \cdot {7.5}^{0.5}\\ t_4 := 545.0353078428827 \cdot {7.5}^{0.5}\\ t_5 := 263.3831869810514 \cdot {7.5}^{0.5}\\ t_6 := {\left({\pi}^{3}\right)}^{0.5}\\ t_7 := 0.1288888888888889 + 0.5 \cdot {t\_0}^{2}\\ t_8 := {2}^{0.5} \cdot \mathsf{fma}\left(t\_5, t\_7, \mathsf{fma}\left(t\_3, t\_0, t\_4\right)\right)\\ t_9 := e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(t\_5, t\_0, t\_3\right)\right)\\ t_10 := \frac{t\_9}{\pi}\\ t_11 := \mathsf{fma}\left(263.3831869810514, t\_2, t\_10\right)\\ \frac{\mathsf{fma}\left(263.3831869810514 \cdot {\pi}^{0.5}, t\_1, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(t\_6, \mathsf{fma}\left(131.6915934905257, t\_2, \frac{\mathsf{fma}\left(e^{-7.5}, t\_8, t\_9\right)}{\pi} - \left(-43.89719783017524 \cdot \pi\right) \cdot t\_1\right), \left(z \cdot \left(\mathsf{fma}\left(0.5, t\_10, \mathsf{fma}\left(43.89719783017524, t\_2, \frac{\mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(t\_5, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, t\_0, 0.16666666666666666 \cdot {t\_0}^{3}\right), \mathsf{fma}\left(t\_3, t\_7, \mathsf{fma}\left(t\_4, t\_0, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot t\_8\right)}{\pi}\right)\right) - \left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right) \cdot t\_11\right)\right) \cdot t\_6\right), t\_6 \cdot t\_11\right)\right)}{z} \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (* -1.0 (log 7.5)) 0.06666666666666667))
        (t_1 (* (exp -7.5) (sqrt 15.0)))
        (t_2 (/ t_1 PI))
        (t_3 (* 436.8961725563396 (pow 7.5 0.5)))
        (t_4 (* 545.0353078428827 (pow 7.5 0.5)))
        (t_5 (* 263.3831869810514 (pow 7.5 0.5)))
        (t_6 (pow (pow PI 3.0) 0.5))
        (t_7 (+ 0.1288888888888889 (* 0.5 (pow t_0 2.0))))
        (t_8 (* (pow 2.0 0.5) (fma t_5 t_7 (fma t_3 t_0 t_4))))
        (t_9 (* (exp -7.5) (* (pow 2.0 0.5) (fma t_5 t_0 t_3))))
        (t_10 (/ t_9 PI))
        (t_11 (fma 263.3831869810514 t_2 t_10)))
   (/
    (fma
     (* 263.3831869810514 (pow PI 0.5))
     t_1
     (*
      z
      (fma
       z
       (fma
        t_6
        (fma
         131.6915934905257
         t_2
         (- (/ (fma (exp -7.5) t_8 t_9) PI) (* (* -43.89719783017524 PI) t_1)))
        (*
         (*
          z
          (-
           (fma
            0.5
            t_10
            (fma
             43.89719783017524
             t_2
             (/
              (fma
               (exp -7.5)
               (*
                (pow 2.0 0.5)
                (fma
                 t_5
                 (+
                  0.008493827160493827
                  (fma
                   0.1288888888888889
                   t_0
                   (* 0.16666666666666666 (pow t_0 3.0))))
                 (fma
                  t_3
                  t_7
                  (fma t_4 t_0 (* 606.6766809167608 (pow 7.5 0.5))))))
               (* (exp -7.5) t_8))
              PI)))
           (* (* -0.16666666666666666 (* PI PI)) t_11)))
         t_6))
       (* t_6 t_11))))
    z)))

C

double code(double z) {
	double t_0 = (-1.0 * log(7.5)) - 0.06666666666666667;
	double t_1 = exp(-7.5) * sqrt(15.0);
	double t_2 = t_1 / ((double) M_PI);
	double t_3 = 436.8961725563396 * pow(7.5, 0.5);
	double t_4 = 545.0353078428827 * pow(7.5, 0.5);
	double t_5 = 263.3831869810514 * pow(7.5, 0.5);
	double t_6 = pow(pow(((double) M_PI), 3.0), 0.5);
	double t_7 = 0.1288888888888889 + (0.5 * pow(t_0, 2.0));
	double t_8 = pow(2.0, 0.5) * fma(t_5, t_7, fma(t_3, t_0, t_4));
	double t_9 = exp(-7.5) * (pow(2.0, 0.5) * fma(t_5, t_0, t_3));
	double t_10 = t_9 / ((double) M_PI);
	double t_11 = fma(263.3831869810514, t_2, t_10);
	return fma((263.3831869810514 * pow(((double) M_PI), 0.5)), t_1, (z * fma(z, fma(t_6, fma(131.6915934905257, t_2, ((fma(exp(-7.5), t_8, t_9) / ((double) M_PI)) - ((-43.89719783017524 * ((double) M_PI)) * t_1))), ((z * (fma(0.5, t_10, fma(43.89719783017524, t_2, (fma(exp(-7.5), (pow(2.0, 0.5) * fma(t_5, (0.008493827160493827 + fma(0.1288888888888889, t_0, (0.16666666666666666 * pow(t_0, 3.0)))), fma(t_3, t_7, fma(t_4, t_0, (606.6766809167608 * pow(7.5, 0.5)))))), (exp(-7.5) * t_8)) / ((double) M_PI)))) - ((-0.16666666666666666 * (((double) M_PI) * ((double) M_PI))) * t_11))) * t_6)), (t_6 * t_11)))) / z;
}

Julia

function code(z)
	t_0 = Float64(Float64(-1.0 * log(7.5)) - 0.06666666666666667)
	t_1 = Float64(exp(-7.5) * sqrt(15.0))
	t_2 = Float64(t_1 / pi)
	t_3 = Float64(436.8961725563396 * (7.5 ^ 0.5))
	t_4 = Float64(545.0353078428827 * (7.5 ^ 0.5))
	t_5 = Float64(263.3831869810514 * (7.5 ^ 0.5))
	t_6 = (pi ^ 3.0) ^ 0.5
	t_7 = Float64(0.1288888888888889 + Float64(0.5 * (t_0 ^ 2.0)))
	t_8 = Float64((2.0 ^ 0.5) * fma(t_5, t_7, fma(t_3, t_0, t_4)))
	t_9 = Float64(exp(-7.5) * Float64((2.0 ^ 0.5) * fma(t_5, t_0, t_3)))
	t_10 = Float64(t_9 / pi)
	t_11 = fma(263.3831869810514, t_2, t_10)
	return Float64(fma(Float64(263.3831869810514 * (pi ^ 0.5)), t_1, Float64(z * fma(z, fma(t_6, fma(131.6915934905257, t_2, Float64(Float64(fma(exp(-7.5), t_8, t_9) / pi) - Float64(Float64(-43.89719783017524 * pi) * t_1))), Float64(Float64(z * Float64(fma(0.5, t_10, fma(43.89719783017524, t_2, Float64(fma(exp(-7.5), Float64((2.0 ^ 0.5) * fma(t_5, Float64(0.008493827160493827 + fma(0.1288888888888889, t_0, Float64(0.16666666666666666 * (t_0 ^ 3.0)))), fma(t_3, t_7, fma(t_4, t_0, Float64(606.6766809167608 * (7.5 ^ 0.5)))))), Float64(exp(-7.5) * t_8)) / pi))) - Float64(Float64(-0.16666666666666666 * Float64(pi * pi)) * t_11))) * t_6)), Float64(t_6 * t_11)))) / z)
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[(-1.0 * N[Log[7.5], $MachinePrecision]), $MachinePrecision] - 0.06666666666666667), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / Pi), $MachinePrecision]}, Block[{t$95$3 = N[(436.8961725563396 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(545.0353078428827 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(263.3831869810514 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Power[Pi, 3.0], $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$7 = N[(0.1288888888888889 + N[(0.5 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$5 * t$95$7 + N[(t$95$3 * t$95$0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$5 * t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$9 / Pi), $MachinePrecision]}, Block[{t$95$11 = N[(263.3831869810514 * t$95$2 + t$95$10), $MachinePrecision]}, N[(N[(N[(263.3831869810514 * N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(z * N[(z * N[(t$95$6 * N[(131.6915934905257 * t$95$2 + N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * t$95$8 + t$95$9), $MachinePrecision] / Pi), $MachinePrecision] - N[(N[(-43.89719783017524 * Pi), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(0.5 * t$95$10 + N[(43.89719783017524 * t$95$2 + N[(N[(N[Exp[-7.5], $MachinePrecision] * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(t$95$5 * N[(0.008493827160493827 + N[(0.1288888888888889 * t$95$0 + N[(0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * t$95$7 + N[(t$95$4 * t$95$0 + N[(606.6766809167608 * N[Power[7.5, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[-7.5], $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Initial program 95.0%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right)\right) + z \cdot \left(z \cdot \left(\sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot \left(\left(\frac{1106209385320415913103082059}{8400000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)}\right)\right) - \frac{-1106209385320415913103082059}{25200000000000000000000000} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right) + \left(z \cdot \left(\left(\frac{1}{2} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)} + \left(\frac{1106209385320415913103082059}{25200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{86}{10125} + \left(\frac{29}{225} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right) + \frac{1}{6} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{3}\right)\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(\frac{29}{225} + \frac{1}{2} \cdot {\left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)}^{2}\right)\right) + \left(\frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{64608921419941589693928044520019}{118540800000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)\right)}{\mathsf{PI}\left(\right)}\right)\right)\right) - \frac{-1}{6} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \cdot \sqrt{{\mathsf{PI}\left(\right)}^{3}}\right) + \sqrt{{\mathsf{PI}\left(\right)}^{3}} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{\mathsf{PI}\left(\right)} + \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\frac{15}{2}} \cdot \left(-1 \cdot \log \frac{15}{2} - \frac{1}{15}\right)\right) + \frac{102757979785251069442117317613}{235200000000000000000000000} \cdot \sqrt{\frac{15}{2}}\right)\right)}{\mathsf{PI}\left(\right)}\right)\right)}{z}} \]

5. Applied rewrites 95.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(263.3831869810514 \cdot {\pi}^{0.5}, e^{-7.5} \cdot \sqrt{15}, z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left({\left({\pi}^{3}\right)}^{0.5}, \mathsf{fma}\left(131.6915934905257, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{\mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)\right)}{\pi} - \left(-43.89719783017524 \cdot \pi\right) \cdot \left(e^{-7.5} \cdot \sqrt{15}\right)\right), \left(z \cdot \left(\mathsf{fma}\left(0.5, \frac{e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)}{\pi}, \mathsf{fma}\left(43.89719783017524, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{\mathsf{fma}\left(e^{-7.5}, {2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.008493827160493827 + \mathsf{fma}\left(0.1288888888888889, -1 \cdot \log 7.5 - 0.06666666666666667, 0.16666666666666666 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{3}\right), \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(545.0353078428827 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 606.6766809167608 \cdot {7.5}^{0.5}\right)\right)\right), e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, 0.1288888888888889 + 0.5 \cdot {\left(-1 \cdot \log 7.5 - 0.06666666666666667\right)}^{2}, \mathsf{fma}\left(436.8961725563396 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 545.0353078428827 \cdot {7.5}^{0.5}\right)\right)\right)\right)}{\pi}\right)\right) - \left(-0.16666666666666666 \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)}{\pi}\right)\right)\right) \cdot {\left({\pi}^{3}\right)}^{0.5}\right), {\left({\pi}^{3}\right)}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514, \frac{e^{-7.5} \cdot \sqrt{15}}{\pi}, \frac{e^{-7.5} \cdot \left({2}^{0.5} \cdot \mathsf{fma}\left(263.3831869810514 \cdot {7.5}^{0.5}, -1 \cdot \log 7.5 - 0.06666666666666667, 436.8961725563396 \cdot {7.5}^{0.5}\right)\right)}{\pi}\right)\right)\right)}{z}} \]
6. Add Preprocessing

Alternative 7: 96.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \pi \cdot 0.5} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (* 263.3831869810514 (/ (* (exp -7.5) (sqrt 15.0)) z))
  (exp (* (log PI) 0.5))))

C

double code(double z) {
	return (263.3831869810514 * ((exp(-7.5) * sqrt(15.0)) / z)) * exp((log(((double) M_PI)) * 0.5));
}

Java

public static double code(double z) {
	return (263.3831869810514 * ((Math.exp(-7.5) * Math.sqrt(15.0)) / z)) * Math.exp((Math.log(Math.PI) * 0.5));
}

Python

def code(z):
	return (263.3831869810514 * ((math.exp(-7.5) * math.sqrt(15.0)) / z)) * math.exp((math.log(math.pi) * 0.5))

Julia

function code(z)
	return Float64(Float64(263.3831869810514 * Float64(Float64(exp(-7.5) * sqrt(15.0)) / z)) * exp(Float64(log(pi) * 0.5)))
end

MATLAB

function tmp = code(z)
	tmp = (263.3831869810514 * ((exp(-7.5) * sqrt(15.0)) / z)) * exp((log(pi) * 0.5));
end

Wolfram

code[z_] := N[(N[(263.3831869810514 * N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Sqrt[15.0], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[Pi], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.0%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \]

5. Applied rewrites 93.8%

   \[\leadsto \color{blue}{\left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{15}}{z}\right) \cdot {\pi}^{0.5}} \]
6. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{2}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z}\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{2}}} \]

   3. pow-to-exp N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}} \]

   4. lower-exp.f64 N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}} \]

   6. lower-log.f64 N/A

      \[\leadsto \left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}} \]

   7. lift-PI.f64 94.1

      \[\leadsto \left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \pi \cdot 0.5} \]

7. Applied rewrites 94.1%

   \[\leadsto \left(263.3831869810514 \cdot \frac{e^{-7.5} \cdot \sqrt{15}}{z}\right) \cdot e^{\log \pi \cdot 0.5} \]
8. Add Preprocessing

Alternative 8: 96.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \pi \cdot 0.5\\ 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot {15}^{0.5}}{z} \cdot \left(\cosh t\_0 + \sinh t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (* (log PI) 0.5)))
   (*
    263.3831869810514
    (* (/ (* (exp -7.5) (pow 15.0 0.5)) z) (+ (cosh t_0) (sinh t_0))))))

C

double code(double z) {
	double t_0 = log(((double) M_PI)) * 0.5;
	return 263.3831869810514 * (((exp(-7.5) * pow(15.0, 0.5)) / z) * (cosh(t_0) + sinh(t_0)));
}

Java

public static double code(double z) {
	double t_0 = Math.log(Math.PI) * 0.5;
	return 263.3831869810514 * (((Math.exp(-7.5) * Math.pow(15.0, 0.5)) / z) * (Math.cosh(t_0) + Math.sinh(t_0)));
}

Python

def code(z):
	t_0 = math.log(math.pi) * 0.5
	return 263.3831869810514 * (((math.exp(-7.5) * math.pow(15.0, 0.5)) / z) * (math.cosh(t_0) + math.sinh(t_0)))

Julia

function code(z)
	t_0 = Float64(log(pi) * 0.5)
	return Float64(263.3831869810514 * Float64(Float64(Float64(exp(-7.5) * (15.0 ^ 0.5)) / z) * Float64(cosh(t_0) + sinh(t_0))))
end

MATLAB

function tmp = code(z)
	t_0 = log(pi) * 0.5;
	tmp = 263.3831869810514 * (((exp(-7.5) * (15.0 ^ 0.5)) / z) * (cosh(t_0) + sinh(t_0)));
end

Wolfram

code[z_] := Block[{t$95$0 = N[(N[Log[Pi], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(263.3831869810514 * N[(N[(N[(N[Exp[-7.5], $MachinePrecision] * N[Power[15.0, 0.5], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(N[Cosh[t$95$0], $MachinePrecision] + N[Sinh[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.0%

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

3. Taylor expanded in z around 0

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(e^{\frac{-15}{2}} + z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)}\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \color{blue}{z} \cdot \left(e^{\frac{-15}{2}} + \frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \left(e^{\frac{-15}{2}} \cdot z + \color{blue}{\left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z}\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, \color{blue}{z}, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot \left(e^{\frac{-15}{2}} + \mathsf{fma}\left(e^{\frac{-15}{2}}, z, \left(\frac{1}{2} \cdot \left(z \cdot e^{\frac{-15}{2}}\right)\right) \cdot z\right)\right)\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
7. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{2 \cdot \frac{15}{2}}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \]

8. Applied rewrites 93.8%

   \[\leadsto \color{blue}{263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot {15}^{0.5}}{z} \cdot {\pi}^{0.5}\right)} \]
9. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{2}}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{2}}}\right) \]

   3. pow-to-exp N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \]

   4. lift-log.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot e^{\log \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot e^{\log \pi \cdot \frac{1}{2}}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot e^{\log \pi \cdot \frac{1}{2}}\right) \]

   7. sinh-+-cosh-rev N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot \left(\cosh \left(\log \pi \cdot \frac{1}{2}\right) + \color{blue}{\sinh \left(\log \pi \cdot \frac{1}{2}\right)}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot \left(\cosh \left(\log \pi \cdot \frac{1}{2}\right) + \color{blue}{\sinh \left(\log \pi \cdot \frac{1}{2}\right)}\right)\right) \]

   9. lower-cosh.f64 N/A

      \[\leadsto \frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot {15}^{\frac{1}{2}}}{z} \cdot \left(\cosh \left(\log \pi \cdot \frac{1}{2}\right) + \sinh \color{blue}{\left(\log \pi \cdot \frac{1}{2}\right)}\right)\right) \]

   10. lower-sinh.f64 94.1

       \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot {15}^{0.5}}{z} \cdot \left(\cosh \left(\log \pi \cdot 0.5\right) + \sinh \left(\log \pi \cdot 0.5\right)\right)\right) \]

10. Applied rewrites 94.1%

    \[\leadsto 263.3831869810514 \cdot \left(\frac{e^{-7.5} \cdot {15}^{0.5}}{z} \cdot \left(\cosh \left(\log \pi \cdot 0.5\right) + \color{blue}{\sinh \left(\log \pi \cdot 0.5\right)}\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))))