Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Percentage Accurate: 93.7% → 96.9%

Time: 15.4s

Alternatives: 9

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Python

def code(x, y, z, t, a, b, c):
	return x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))

Julia

function code(x, y, z, t, a, b, c)
	return Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
}

Python

def code(x, y, z, t, a, b, c):
	return x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))

Julia

function code(x, y, z, t, a, b, c)
	return Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{e^{t\_1 \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{a}}, \frac{z}{b \cdot t}, \frac{c}{b} - 1\right) \cdot a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* (sqrt (+ a t)) z) t)
          (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))))
   (if (<= t_1 INFINITY)
     (/ x (+ (* (exp (* t_1 2.0)) y) x))
     (/
      x
      (+
       (*
        (exp
         (*
          (* (* (fma (sqrt (/ 1.0 a)) (/ z (* b t)) (- (/ c b) 1.0)) a) b)
          2.0))
        y)
       x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = x / ((exp((t_1 * 2.0)) * y) + x);
	} else {
		tmp = x / ((exp((((fma(sqrt((1.0 / a)), (z / (b * t)), ((c / b) - 1.0)) * a) * b) * 2.0)) * y) + x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(x / Float64(Float64(exp(Float64(t_1 * 2.0)) * y) + x));
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(fma(sqrt(Float64(1.0 / a)), Float64(z / Float64(b * t)), Float64(Float64(c / b) - 1.0)) * a) * b) * 2.0)) * y) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(x / N[(N[(N[Exp[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[N[(N[(N[(N[(N[Sqrt[N[(1.0 / a), $MachinePrecision]], $MachinePrecision] * N[(z / N[(b * t), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < +inf.0

   1. Initial program 99.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   if +inf.0 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 0.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\left(\frac{2}{3} \cdot \frac{1}{t} + \frac{z}{b \cdot t} \cdot \sqrt{a + t}\right) - \left(\frac{5}{6} + \left(a + -1 \cdot \frac{c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}{b}\right)\right)\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{2}{3} \cdot \frac{1}{t} + \frac{z}{b \cdot t} \cdot \sqrt{a + t}\right) - \left(\frac{5}{6} + \left(a + -1 \cdot \frac{c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}{b}\right)\right)\right) \cdot b\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{2}{3} \cdot \frac{1}{t} + \frac{z}{b \cdot t} \cdot \sqrt{a + t}\right) - \left(\frac{5}{6} + \left(a + -1 \cdot \frac{c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}{b}\right)\right)\right) \cdot b\right)}}} \]

   5. Applied rewrites 44.2%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\frac{\frac{z}{t}}{b}, \sqrt{a + t}, \frac{0.6666666666666666}{t}\right) - \left(\left(0.8333333333333334 + a\right) - \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \frac{c}{b}\right)\right) \cdot b\right)}}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot \left(\left(\sqrt{\frac{1}{a}} \cdot \frac{z}{b \cdot t} + \frac{c}{b}\right) - 1\right)\right) \cdot b\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{a}}, \frac{z}{b \cdot t}, \frac{c}{b} - 1\right) \cdot a\right) \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq \infty:\\ \;\;\;\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{a}}, \frac{z}{b \cdot t}, \frac{c}{b} - 1\right) \cdot a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x} \leq 10^{-69}:\\ \;\;\;\;\frac{x}{e^{\left(\left(-b\right) \cdot a\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        (*
         (exp
          (*
           (-
            (/ (* (sqrt (+ a t)) z) t)
            (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))
           2.0))
         y)
        x))
      1e-69)
   (/ x (+ (* (exp (* (* (- b) a) 2.0)) y) x))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / ((exp(((((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69) {
		tmp = x / ((exp(((-b * a) * 2.0)) * y) + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / ((exp(((((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))) * 2.0d0)) * y) + x)) <= 1d-69) then
        tmp = x / ((exp(((-b * a) * 2.0d0)) * y) + x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / ((Math.exp(((((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69) {
		tmp = x / ((Math.exp(((-b * a) * 2.0)) * y) + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / ((math.exp(((((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69:
		tmp = x / ((math.exp(((-b * a) * 2.0)) * y) + x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c))) * 2.0)) * y) + x)) <= 1e-69)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(-b) * a) * 2.0)) * y) + x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / ((exp(((((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69)
		tmp = x / ((exp(((-b * a) * 2.0)) * y) + x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(N[(N[Exp[N[(N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1e-69], N[(x / N[(N[(N[Exp[N[(N[((-b) * a), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 9.9999999999999996e-70

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      3. associate--r+ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right)} - a\right) \cdot b\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{t}} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\frac{\color{blue}{\frac{2}{3}}}{t} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      8. lower-/.f64 70.7

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{0.6666666666666666}{t}} - 0.8333333333333334\right) - a\right) \cdot b\right)}} \]

   5. Applied rewrites 70.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right)}}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(-1 \cdot \color{blue}{\left(a \cdot b\right)}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(-b\right) \cdot \color{blue}{a}\right)}} \]

   if 9.9999999999999996e-70 < (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))))))

   1. Initial program 83.5%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.2

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.2%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x} \leq 10^{-69}:\\ \;\;\;\;\frac{x}{e^{\left(\left(-b\right) \cdot a\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x} \leq 10^{-69}:\\ \;\;\;\;\frac{x}{e^{\left(c \cdot a\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        (*
         (exp
          (*
           (-
            (/ (* (sqrt (+ a t)) z) t)
            (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))
           2.0))
         y)
        x))
      1e-69)
   (/ x (+ (* (exp (* (* c a) 2.0)) y) x))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / ((exp(((((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69) {
		tmp = x / ((exp(((c * a) * 2.0)) * y) + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / ((exp(((((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))) * 2.0d0)) * y) + x)) <= 1d-69) then
        tmp = x / ((exp(((c * a) * 2.0d0)) * y) + x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / ((Math.exp(((((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69) {
		tmp = x / ((Math.exp(((c * a) * 2.0)) * y) + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / ((math.exp(((((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69:
		tmp = x / ((math.exp(((c * a) * 2.0)) * y) + x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c))) * 2.0)) * y) + x)) <= 1e-69)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(c * a) * 2.0)) * y) + x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / ((exp(((((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) * 2.0)) * y) + x)) <= 1e-69)
		tmp = x / ((exp(((c * a) * 2.0)) * y) + x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(N[(N[Exp[N[(N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1e-69], N[(x / N[(N[(N[Exp[N[(N[(c * a), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 9.9999999999999996e-70

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 60.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \color{blue}{c}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{a}\right)}} \]

   if 9.9999999999999996e-70 < (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))))))

   1. Initial program 83.5%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.2

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.2%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x} \leq 10^{-69}:\\ \;\;\;\;\frac{x}{e^{\left(c \cdot a\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* (sqrt (+ a t)) z) t)
          (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))))
   (if (<= t_1 -20000000000.0)
     1.0
     (if (<= t_1 4e+121)
       (/
        x
        (+
         (*
          (exp
           (* (* (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)) c) 2.0))
          y)
         x))
       (/
        x
        (+
         (*
          (exp
           (* (* (- (- (/ 0.6666666666666666 t) 0.8333333333333334) a) b) 2.0))
          y)
         x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+121) {
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))
    if (t_1 <= (-20000000000.0d0)) then
        tmp = 1.0d0
    else if (t_1 <= 4d+121) then
        tmp = x / ((exp(((((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t)) * c) * 2.0d0)) * y) + x)
    else
        tmp = x / ((exp((((((0.6666666666666666d0 / t) - 0.8333333333333334d0) - a) * b) * 2.0d0)) * y) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+121) {
		tmp = x / ((Math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((Math.exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = 1.0
	elif t_1 <= 4e+121:
		tmp = x / ((math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x)
	else:
		tmp = x / ((math.exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c)))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+121)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)) * c) * 2.0)) * y) + x));
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(Float64(0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+121)
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	else
		tmp = x / ((exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], 1.0, If[LessEqual[t$95$1, 4e+121], N[(x / N[(N[(N[Exp[N[(N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[N[(N[(N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - 0.8333333333333334), $MachinePrecision] - a), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e10

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} \]

   if -2e10 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 4.00000000000000015e121

   1. Initial program 100.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 88.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 88.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   if 4.00000000000000015e121 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 78.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      3. associate--r+ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right)} - a\right) \cdot b\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{t}} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\frac{\color{blue}{\frac{2}{3}}}{t} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      8. lower-/.f64 70.8

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{0.6666666666666666}{t}} - 0.8333333333333334\right) - a\right) \cdot b\right)}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq 4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+289}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{b}{t} \cdot 0.6666666666666666\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* (sqrt (+ a t)) z) t)
          (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))))
   (if (<= t_1 -20000000000.0)
     1.0
     (if (<= t_1 1e+289)
       (/ x (+ (* (exp (* (* (+ 0.8333333333333334 a) c) 2.0)) y) x))
       (/ x (+ (* (exp (* (* (/ b t) 0.6666666666666666) 2.0)) y) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 1e+289) {
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((exp((((b / t) * 0.6666666666666666) * 2.0)) * y) + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))
    if (t_1 <= (-20000000000.0d0)) then
        tmp = 1.0d0
    else if (t_1 <= 1d+289) then
        tmp = x / ((exp((((0.8333333333333334d0 + a) * c) * 2.0d0)) * y) + x)
    else
        tmp = x / ((exp((((b / t) * 0.6666666666666666d0) * 2.0d0)) * y) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 1e+289) {
		tmp = x / ((Math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((Math.exp((((b / t) * 0.6666666666666666) * 2.0)) * y) + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = 1.0
	elif t_1 <= 1e+289:
		tmp = x / ((math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x)
	else:
		tmp = x / ((math.exp((((b / t) * 0.6666666666666666) * 2.0)) * y) + x)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c)))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 1e+289)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(0.8333333333333334 + a) * c) * 2.0)) * y) + x));
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(b / t) * 0.6666666666666666) * 2.0)) * y) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 1e+289)
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	else
		tmp = x / ((exp((((b / t) * 0.6666666666666666) * 2.0)) * y) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], 1.0, If[LessEqual[t$95$1, 1e+289], N[(x / N[(N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[N[(N[(N[(b / t), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e10

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} \]

   if -2e10 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 1.0000000000000001e289

   1. Initial program 100.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 73.7

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 73.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{5}{6} + \color{blue}{a}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.4%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + \color{blue}{a}\right)\right)}} \]

   if 1.0000000000000001e289 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 69.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      3. associate--r+ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right)} - a\right) \cdot b\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{t}} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\frac{\color{blue}{\frac{2}{3}}}{t} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      8. lower-/.f64 72.7

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{0.6666666666666666}{t}} - 0.8333333333333334\right) - a\right) \cdot b\right)}} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right)}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{2}{3} \cdot \color{blue}{\frac{b}{t}}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{b}{t} \cdot \color{blue}{0.6666666666666666}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq 10^{+289}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{b}{t} \cdot 0.6666666666666666\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(-0.8333333333333334 - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* (sqrt (+ a t)) z) t)
          (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))))
   (if (<= t_1 -20000000000.0)
     1.0
     (if (<= t_1 4e+121)
       (/ x (+ (* (exp (* (* (+ 0.8333333333333334 a) c) 2.0)) y) x))
       (/ x (+ (* (exp (* (* (- -0.8333333333333334 a) b) 2.0)) y) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+121) {
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((exp((((-0.8333333333333334 - a) * b) * 2.0)) * y) + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))
    if (t_1 <= (-20000000000.0d0)) then
        tmp = 1.0d0
    else if (t_1 <= 4d+121) then
        tmp = x / ((exp((((0.8333333333333334d0 + a) * c) * 2.0d0)) * y) + x)
    else
        tmp = x / ((exp(((((-0.8333333333333334d0) - a) * b) * 2.0d0)) * y) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+121) {
		tmp = x / ((Math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((Math.exp((((-0.8333333333333334 - a) * b) * 2.0)) * y) + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = 1.0
	elif t_1 <= 4e+121:
		tmp = x / ((math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x)
	else:
		tmp = x / ((math.exp((((-0.8333333333333334 - a) * b) * 2.0)) * y) + x)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c)))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+121)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(0.8333333333333334 + a) * c) * 2.0)) * y) + x));
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(-0.8333333333333334 - a) * b) * 2.0)) * y) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+121)
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	else
		tmp = x / ((exp((((-0.8333333333333334 - a) * b) * 2.0)) * y) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], 1.0, If[LessEqual[t$95$1, 4e+121], N[(x / N[(N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[N[(N[(N[(-0.8333333333333334 - a), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e10

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} \]

   if -2e10 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 4.00000000000000015e121

   1. Initial program 100.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 88.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 88.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{5}{6} + \color{blue}{a}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.2%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + \color{blue}{a}\right)\right)}} \]

   if 4.00000000000000015e121 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 78.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      3. associate--r+ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right)} - a\right) \cdot b\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{t}} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\frac{\color{blue}{\frac{2}{3}}}{t} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      8. lower-/.f64 70.8

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{0.6666666666666666}{t}} - 0.8333333333333334\right) - a\right) \cdot b\right)}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right)}}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\frac{-5}{6} - a\right) \cdot b\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(-0.8333333333333334 - a\right) \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq 4 \cdot 10^{+121}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(-0.8333333333333334 - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right)\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+289}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(-b\right) \cdot a\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* (sqrt (+ a t)) z) t)
          (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))))
   (if (<= t_1 -20000000000.0)
     1.0
     (if (<= t_1 1e+289)
       (/ x (+ (* (exp (* (* (+ 0.8333333333333334 a) c) 2.0)) y) x))
       (/ x (+ (* (exp (* (* (- b) a) 2.0)) y) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 1e+289) {
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((exp(((-b * a) * 2.0)) * y) + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))
    if (t_1 <= (-20000000000.0d0)) then
        tmp = 1.0d0
    else if (t_1 <= 1d+289) then
        tmp = x / ((exp((((0.8333333333333334d0 + a) * c) * 2.0d0)) * y) + x)
    else
        tmp = x / ((exp(((-b * a) * 2.0d0)) * y) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 1e+289) {
		tmp = x / ((Math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	} else {
		tmp = x / ((Math.exp(((-b * a) * 2.0)) * y) + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = 1.0
	elif t_1 <= 1e+289:
		tmp = x / ((math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x)
	else:
		tmp = x / ((math.exp(((-b * a) * 2.0)) * y) + x)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c)))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 1e+289)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(0.8333333333333334 + a) * c) * 2.0)) * y) + x));
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(-b) * a) * 2.0)) * y) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c));
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = 1.0;
	elseif (t_1 <= 1e+289)
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	else
		tmp = x / ((exp(((-b * a) * 2.0)) * y) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], 1.0, If[LessEqual[t$95$1, 1e+289], N[(x / N[(N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[N[(N[((-b) * a), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e10

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} \]

   if -2e10 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 1.0000000000000001e289

   1. Initial program 100.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 73.7

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 73.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\frac{5}{6} + \color{blue}{a}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.4%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + \color{blue}{a}\right)\right)}} \]

   if 1.0000000000000001e289 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 69.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(b \cdot \left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right)\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \left(\frac{5}{6} + a\right)\right) \cdot b\right)}}} \]

      3. associate--r+ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right) - a\right)} \cdot b\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\color{blue}{\left(\frac{2}{3} \cdot \frac{1}{t} - \frac{5}{6}\right)} - a\right) \cdot b\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{t}} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\frac{\color{blue}{\frac{2}{3}}}{t} - \frac{5}{6}\right) - a\right) \cdot b\right)}} \]

      8. lower-/.f64 72.7

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\color{blue}{\frac{0.6666666666666666}{t}} - 0.8333333333333334\right) - a\right) \cdot b\right)}} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right)}}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(-1 \cdot \color{blue}{\left(a \cdot b\right)}\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.3%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\left(-b\right) \cdot \color{blue}{a}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq 10^{+289}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(-b\right) \cdot a\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (-
       (/ (* (sqrt (+ a t)) z) t)
       (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* 3.0 t))) (- b c)))
      -20000000000.0)
   1.0
   (/
    x
    (+
     (*
      (exp (* (* (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)) c) 2.0))
      y)
     x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) <= -20000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((((sqrt((a + t)) * z) / t) - ((((5.0d0 / 6.0d0) + a) - (2.0d0 / (3.0d0 * t))) * (b - c))) <= (-20000000000.0d0)) then
        tmp = 1.0d0
    else
        tmp = x / ((exp(((((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t)) * c) * 2.0d0)) * y) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((Math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) <= -20000000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / ((Math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (((math.sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) <= -20000000000.0:
		tmp = 1.0
	else:
		tmp = x / ((math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(Float64(5.0 / 6.0) + a) - Float64(2.0 / Float64(3.0 * t))) * Float64(b - c))) <= -20000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)) * c) * 2.0)) * y) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((((sqrt((a + t)) * z) / t) - ((((5.0 / 6.0) + a) - (2.0 / (3.0 * t))) * (b - c))) <= -20000000000.0)
		tmp = 1.0;
	else
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] - N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -20000000000.0], 1.0, N[(x / N[(N[(N[Exp[N[(N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e10

   1. Initial program 99.1%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 76.9

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 76.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{1} \]

   if -2e10 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 83.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

      6. lower-/.f64 63.8

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

   5. Applied rewrites 63.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{3 \cdot t}\right) \cdot \left(b - c\right) \leq -20000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.2% accurate, 198.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 1.0)

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return 1.0;
}

Python

def code(x, y, z, t, a, b, c):
	return 1.0

Julia

function code(x, y, z, t, a, b, c)
	return 1.0
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := 1.0

Derivation

1. Initial program 90.3%

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{\left(\left(\frac{5}{6} + a\right) - \frac{2}{3} \cdot \frac{1}{t}\right)}\right)}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\color{blue}{\left(\frac{5}{6} + a\right)} - \frac{2}{3} \cdot \frac{1}{t}\right)\right)}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \color{blue}{\frac{\frac{2}{3} \cdot 1}{t}}\right)\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(\frac{5}{6} + a\right) - \frac{\color{blue}{\frac{2}{3}}}{t}\right)\right)}} \]

   6. lower-/.f64 69.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + a\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)}} \]

5. Applied rewrites 69.5%

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)\right)}}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 54.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \sqrt{t + a}\\ t_2 := a - \frac{5}{6}\\ \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{t\_1 \cdot \left(\left(3 \cdot t\right) \cdot t\_2\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(t\_2 \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot t\_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{t\_1}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* z (sqrt (+ t a)))) (t_2 (- a (/ 5.0 6.0))))
   (if (< t -2.118326644891581e-50)
     (/
      x
      (+
       x
       (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b)))))))
     (if (< t 5.196588770651547e-123)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (/
             (-
              (* t_1 (* (* 3.0 t) t_2))
              (*
               (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0)
               (* t_2 (* (- b c) t))))
             (* (* (* t t) 3.0) t_2)))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (/ t_1 t)
             (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = z * sqrt((t + a));
	double t_2 = a - (5.0 / 6.0);
	double tmp;
	if (t < -2.118326644891581e-50) {
		tmp = x / (x + (y * exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	} else if (t < 5.196588770651547e-123) {
		tmp = x / (x + (y * exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = z * sqrt((t + a))
    t_2 = a - (5.0d0 / 6.0d0)
    if (t < (-2.118326644891581d-50)) then
        tmp = x / (x + (y * exp((2.0d0 * (((a * c) + (0.8333333333333334d0 * c)) - (a * b))))))
    else if (t < 5.196588770651547d-123) then
        tmp = x / (x + (y * exp((2.0d0 * (((t_1 * ((3.0d0 * t) * t_2)) - (((((5.0d0 / 6.0d0) + a) * (3.0d0 * t)) - 2.0d0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0d0) * t_2))))))
    else
        tmp = x / (x + (y * exp((2.0d0 * ((t_1 / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = z * Math.sqrt((t + a));
	double t_2 = a - (5.0 / 6.0);
	double tmp;
	if (t < -2.118326644891581e-50) {
		tmp = x / (x + (y * Math.exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	} else if (t < 5.196588770651547e-123) {
		tmp = x / (x + (y * Math.exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = z * math.sqrt((t + a))
	t_2 = a - (5.0 / 6.0)
	tmp = 0
	if t < -2.118326644891581e-50:
		tmp = x / (x + (y * math.exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))))
	elif t < 5.196588770651547e-123:
		tmp = x / (x + (y * math.exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(z * sqrt(Float64(t + a)))
	t_2 = Float64(a - Float64(5.0 / 6.0))
	tmp = 0.0
	if (t < -2.118326644891581e-50)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(a * c) + Float64(0.8333333333333334 * c)) - Float64(a * b)))))));
	elseif (t < 5.196588770651547e-123)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(t_1 * Float64(Float64(3.0 * t) * t_2)) - Float64(Float64(Float64(Float64(Float64(5.0 / 6.0) + a) * Float64(3.0 * t)) - 2.0) * Float64(t_2 * Float64(Float64(b - c) * t)))) / Float64(Float64(Float64(t * t) * 3.0) * t_2)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(t_1 / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = z * sqrt((t + a));
	t_2 = a - (5.0 / 6.0);
	tmp = 0.0;
	if (t < -2.118326644891581e-50)
		tmp = x / (x + (y * exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	elseif (t < 5.196588770651547e-123)
		tmp = x / (x + (y * exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	else
		tmp = x / (x + (y * exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a - N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -2.118326644891581e-50], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(a * c), $MachinePrecision] + N[(0.8333333333333334 * c), $MachinePrecision]), $MachinePrecision] - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[t, 5.196588770651547e-123], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(t$95$1 * N[(N[(3.0 * t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision] * N[(3.0 * t), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * N[(t$95$2 * N[(N[(b - c), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t * t), $MachinePrecision] * 3.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(t$95$1 / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024270 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -2118326644891581/100000000000000000000000000000000000000000000000000000000000000000) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 4166666666666667/5000000000000000 c)) (* a b))))))) (if (< t 5196588770651547/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))