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

Percentage Accurate: 94.2% → 97.3%

Time: 12.9s

Alternatives: 11

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: 94.2% 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: 97.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 INFINITY)
     (/ x (+ x (* y (exp (* 2.0 t_1)))))
     (/
      x
      (+
       x
       (*
        y
        (exp
         (* 2.0 (/ (fma 0.6666666666666666 (- b c) (* (sqrt a) z)) t)))))))))

C

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

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * t_1)))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(0.6666666666666666, Float64(b - c), Float64(sqrt(a) * z)) / t))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(0.6666666666666666 * N[(b - c), $MachinePrecision] + N[(N[Sqrt[a], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 t around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\mathsf{fma}\left(0.6666666666666666, b - c, \sqrt{a} \cdot z\right)}{t}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.8333333333333334 + a\right) \cdot c\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(-b\right) \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+28)
     1.0
     (if (<= t_1 5e+200)
       (/ x (+ x (* y (exp (* 2.0 (* (+ 0.8333333333333334 a) c))))))
       (if (<= t_1 1e+300)
         (/ x (+ x (* y (exp (* 2.0 (* (- b) (+ 0.8333333333333334 a)))))))
         (/ x (+ x (* y (exp (* 2.0 (* c a)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 5e+200) {
		tmp = x / (x + (y * exp((2.0 * ((0.8333333333333334 + a) * c)))));
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * exp((2.0 * (-b * (0.8333333333333334 + a))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+28)) then
        tmp = 1.0d0
    else if (t_1 <= 5d+200) then
        tmp = x / (x + (y * exp((2.0d0 * ((0.8333333333333334d0 + a) * c)))))
    else if (t_1 <= 1d+300) then
        tmp = x / (x + (y * exp((2.0d0 * (-b * (0.8333333333333334d0 + a))))))
    else
        tmp = x / (x + (y * exp((2.0d0 * (c * a)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 5e+200) {
		tmp = x / (x + (y * Math.exp((2.0 * ((0.8333333333333334 + a) * c)))));
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * Math.exp((2.0 * (-b * (0.8333333333333334 + a))))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (c * a)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+28:
		tmp = 1.0
	elif t_1 <= 5e+200:
		tmp = x / (x + (y * math.exp((2.0 * ((0.8333333333333334 + a) * c)))))
	elif t_1 <= 1e+300:
		tmp = x / (x + (y * math.exp((2.0 * (-b * (0.8333333333333334 + a))))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (c * a)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 5e+200)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(0.8333333333333334 + a) * c))))));
	elseif (t_1 <= 1e+300)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(-b) * Float64(0.8333333333333334 + a)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 5e+200)
		tmp = x / (x + (y * exp((2.0 * ((0.8333333333333334 + a) * c)))));
	elseif (t_1 <= 1e+300)
		tmp = x / (x + (y * exp((2.0 * (-b * (0.8333333333333334 + a))))));
	else
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+28], 1.0, If[LessEqual[t$95$1, 5e+200], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[((-b) * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))))) < -9.99999999999999958e27

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.99999999999999958e27 < (-.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)))))) < 5.00000000000000019e200

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 71.0%

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

   if 5.00000000000000019e200 < (-.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)))))) < 1.0000000000000001e300

   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 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. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 63.0%

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

   if 1.0000000000000001e300 < (-.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 75.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   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 56.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{a}\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.8333333333333334 + a\right) \cdot c\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-0.8333333333333334 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+28)
     1.0
     (if (<= t_1 4e+212)
       (/ x (+ x (* y (exp (* 2.0 (* (+ 0.8333333333333334 a) c))))))
       (if (<= t_1 1e+300)
         (/ x (+ x (* y (exp (* 2.0 (* -0.8333333333333334 b))))))
         (/ x (+ x (* y (exp (* 2.0 (* c a)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 4e+212) {
		tmp = x / (x + (y * exp((2.0 * ((0.8333333333333334 + a) * c)))));
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * exp((2.0 * (-0.8333333333333334 * b)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+28)) then
        tmp = 1.0d0
    else if (t_1 <= 4d+212) then
        tmp = x / (x + (y * exp((2.0d0 * ((0.8333333333333334d0 + a) * c)))))
    else if (t_1 <= 1d+300) then
        tmp = x / (x + (y * exp((2.0d0 * ((-0.8333333333333334d0) * b)))))
    else
        tmp = x / (x + (y * exp((2.0d0 * (c * a)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 4e+212) {
		tmp = x / (x + (y * Math.exp((2.0 * ((0.8333333333333334 + a) * c)))));
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * Math.exp((2.0 * (-0.8333333333333334 * b)))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (c * a)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+28:
		tmp = 1.0
	elif t_1 <= 4e+212:
		tmp = x / (x + (y * math.exp((2.0 * ((0.8333333333333334 + a) * c)))))
	elif t_1 <= 1e+300:
		tmp = x / (x + (y * math.exp((2.0 * (-0.8333333333333334 * b)))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (c * a)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 4e+212)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(0.8333333333333334 + a) * c))))));
	elseif (t_1 <= 1e+300)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(-0.8333333333333334 * b))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 4e+212)
		tmp = x / (x + (y * exp((2.0 * ((0.8333333333333334 + a) * c)))));
	elseif (t_1 <= 1e+300)
		tmp = x / (x + (y * exp((2.0 * (-0.8333333333333334 * b)))));
	else
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+28], 1.0, If[LessEqual[t$95$1, 4e+212], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(-0.8333333333333334 * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))))) < -9.99999999999999958e27

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.99999999999999958e27 < (-.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)))))) < 3.9999999999999996e212

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.1%

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

   if 3.9999999999999996e212 < (-.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)))))) < 1.0000000000000001e300

   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 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. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 65.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 62.1%

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

   if 1.0000000000000001e300 < (-.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 75.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   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 56.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{a}\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.8333333333333334 \cdot c\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-0.8333333333333334 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+24)
     1.0
     (if (<= t_1 5e+200)
       (/ x (+ x (* y (exp (* 2.0 (* 0.8333333333333334 c))))))
       (if (<= t_1 1e+300)
         (/ x (+ x (* y (exp (* 2.0 (* -0.8333333333333334 b))))))
         (/ x (+ x (* y (exp (* 2.0 (* c a)))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = 1.0;
	} else if (t_1 <= 5e+200) {
		tmp = x / (x + (y * exp((2.0 * (0.8333333333333334 * c)))));
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * exp((2.0 * (-0.8333333333333334 * b)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+24)) then
        tmp = 1.0d0
    else if (t_1 <= 5d+200) then
        tmp = x / (x + (y * exp((2.0d0 * (0.8333333333333334d0 * c)))))
    else if (t_1 <= 1d+300) then
        tmp = x / (x + (y * exp((2.0d0 * ((-0.8333333333333334d0) * b)))))
    else
        tmp = x / (x + (y * exp((2.0d0 * (c * a)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = 1.0;
	} else if (t_1 <= 5e+200) {
		tmp = x / (x + (y * Math.exp((2.0 * (0.8333333333333334 * c)))));
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * Math.exp((2.0 * (-0.8333333333333334 * b)))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (c * a)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+24:
		tmp = 1.0
	elif t_1 <= 5e+200:
		tmp = x / (x + (y * math.exp((2.0 * (0.8333333333333334 * c)))))
	elif t_1 <= 1e+300:
		tmp = x / (x + (y * math.exp((2.0 * (-0.8333333333333334 * b)))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (c * a)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+24)
		tmp = 1.0;
	elseif (t_1 <= 5e+200)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(0.8333333333333334 * c))))));
	elseif (t_1 <= 1e+300)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(-0.8333333333333334 * b))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+24)
		tmp = 1.0;
	elseif (t_1 <= 5e+200)
		tmp = x / (x + (y * exp((2.0 * (0.8333333333333334 * c)))));
	elseif (t_1 <= 1e+300)
		tmp = x / (x + (y * exp((2.0 * (-0.8333333333333334 * b)))));
	else
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+24], 1.0, If[LessEqual[t$95$1, 5e+200], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(0.8333333333333334 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+300], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(-0.8333333333333334 * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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)))))) < -9.9999999999999998e23

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.9999999999999998e23 < (-.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)))))) < 5.00000000000000019e200

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 60.8%

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

   if 5.00000000000000019e200 < (-.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)))))) < 1.0000000000000001e300

   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 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. +-commutative N/A

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

      4. associate--r+ N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 58.0%

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

   if 1.0000000000000001e300 < (-.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 75.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   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 56.8%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \color{blue}{a}\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right) \cdot b\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+28)
     1.0
     (if (<= t_1 5e+200)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)) c))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (*
             (- (- (/ 0.6666666666666666 t) a) 0.8333333333333334)
             b))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 5e+200) {
		tmp = x / (x + (y * exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * ((((0.6666666666666666 / t) - a) - 0.8333333333333334) * b)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+28)) then
        tmp = 1.0d0
    else if (t_1 <= 5d+200) then
        tmp = x / (x + (y * exp((2.0d0 * (((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t)) * c)))))
    else
        tmp = x / (x + (y * exp((2.0d0 * ((((0.6666666666666666d0 / t) - a) - 0.8333333333333334d0) * b)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 5e+200) {
		tmp = x / (x + (y * Math.exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * ((((0.6666666666666666 / t) - a) - 0.8333333333333334) * b)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+28:
		tmp = 1.0
	elif t_1 <= 5e+200:
		tmp = x / (x + (y * math.exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * ((((0.6666666666666666 / t) - a) - 0.8333333333333334) * b)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 5e+200)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)) * c))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(Float64(0.6666666666666666 / t) - a) - 0.8333333333333334) * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 5e+200)
		tmp = x / (x + (y * exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))));
	else
		tmp = x / (x + (y * exp((2.0 * ((((0.6666666666666666 / t) - a) - 0.8333333333333334) * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+28], 1.0, If[LessEqual[t$95$1, 5e+200], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - a), $MachinePrecision] - 0.8333333333333334), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)))))) < -9.99999999999999958e27

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.99999999999999958e27 < (-.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)))))) < 5.00000000000000019e200

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if 5.00000000000000019e200 < (-.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 85.9%

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

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

      4. associate--r+ N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

Alternative 6: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) \cdot b\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+28)
     1.0
     (if (<= t_1 5e+215)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)) c))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* (- (/ 0.6666666666666666 t) 0.8333333333333334) b))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 5e+215) {
		tmp = x / (x + (y * exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+28)) then
        tmp = 1.0d0
    else if (t_1 <= 5d+215) then
        tmp = x / (x + (y * exp((2.0d0 * (((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t)) * c)))))
    else
        tmp = x / (x + (y * exp((2.0d0 * (((0.6666666666666666d0 / t) - 0.8333333333333334d0) * b)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 5e+215) {
		tmp = x / (x + (y * Math.exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+28:
		tmp = 1.0
	elif t_1 <= 5e+215:
		tmp = x / (x + (y * math.exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 5e+215)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)) * c))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(0.6666666666666666 / t) - 0.8333333333333334) * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 5e+215)
		tmp = x / (x + (y * exp((2.0 * (((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c)))));
	else
		tmp = x / (x + (y * exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+28], 1.0, If[LessEqual[t$95$1, 5e+215], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - 0.8333333333333334), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)))))) < -9.99999999999999958e27

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.99999999999999958e27 < (-.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)))))) < 5.0000000000000001e215

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if 5.0000000000000001e215 < (-.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 84.7%

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

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

      4. associate--r+ N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.8%

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

Alternative 7: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(0.8333333333333334 + a\right) \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) \cdot b\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+28)
     1.0
     (if (<= t_1 4e+212)
       (/ x (+ x (* y (exp (* 2.0 (* (+ 0.8333333333333334 a) c))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* (- (/ 0.6666666666666666 t) 0.8333333333333334) b))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 4e+212) {
		tmp = x / (x + (y * exp((2.0 * ((0.8333333333333334 + a) * c)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+28)) then
        tmp = 1.0d0
    else if (t_1 <= 4d+212) then
        tmp = x / (x + (y * exp((2.0d0 * ((0.8333333333333334d0 + a) * c)))))
    else
        tmp = x / (x + (y * exp((2.0d0 * (((0.6666666666666666d0 / t) - 0.8333333333333334d0) * b)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+28) {
		tmp = 1.0;
	} else if (t_1 <= 4e+212) {
		tmp = x / (x + (y * Math.exp((2.0 * ((0.8333333333333334 + a) * c)))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+28:
		tmp = 1.0
	elif t_1 <= 4e+212:
		tmp = x / (x + (y * math.exp((2.0 * ((0.8333333333333334 + a) * c)))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 4e+212)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(0.8333333333333334 + a) * c))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(0.6666666666666666 / t) - 0.8333333333333334) * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+28)
		tmp = 1.0;
	elseif (t_1 <= 4e+212)
		tmp = x / (x + (y * exp((2.0 * ((0.8333333333333334 + a) * c)))));
	else
		tmp = x / (x + (y * exp((2.0 * (((0.6666666666666666 / t) - 0.8333333333333334) * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+28], 1.0, If[LessEqual[t$95$1, 4e+212], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - 0.8333333333333334), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)))))) < -9.99999999999999958e27

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.99999999999999958e27 < (-.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)))))) < 3.9999999999999996e212

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.1%

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

   if 3.9999999999999996e212 < (-.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 84.9%

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

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

      4. associate--r+ N/A

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

      5. unsub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.1%

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

Alternative 8: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+300}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.8333333333333334 \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (-
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))
   (if (<= t_1 -1e+24)
     1.0
     (if (<= t_1 1e+300)
       (/ x (+ x (* y (exp (* 2.0 (* 0.8333333333333334 c))))))
       (/ x (+ x (* y (exp (* 2.0 (* c a))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = 1.0;
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * exp((2.0 * (0.8333333333333334 * c)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	}
	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 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))
    if (t_1 <= (-1d+24)) then
        tmp = 1.0d0
    else if (t_1 <= 1d+300) then
        tmp = x / (x + (y * exp((2.0d0 * (0.8333333333333334d0 * c)))))
    else
        tmp = x / (x + (y * exp((2.0d0 * (c * a)))))
    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))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = 1.0;
	} else if (t_1 <= 1e+300) {
		tmp = x / (x + (y * Math.exp((2.0 * (0.8333333333333334 * c)))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (c * a)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))
	tmp = 0
	if t_1 <= -1e+24:
		tmp = 1.0
	elif t_1 <= 1e+300:
		tmp = x / (x + (y * math.exp((2.0 * (0.8333333333333334 * c)))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (c * a)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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)))))
	tmp = 0.0
	if (t_1 <= -1e+24)
		tmp = 1.0;
	elseif (t_1 <= 1e+300)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(0.8333333333333334 * c))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))));
	tmp = 0.0;
	if (t_1 <= -1e+24)
		tmp = 1.0;
	elseif (t_1 <= 1e+300)
		tmp = x / (x + (y * exp((2.0 * (0.8333333333333334 * c)))));
	else
		tmp = x / (x + (y * exp((2.0 * (c * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+24], 1.0, If[LessEqual[t$95$1, 1e+300], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(0.8333333333333334 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)))))) < -9.9999999999999998e23

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.9999999999999998e23 < (-.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)))))) < 1.0000000000000001e300

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 57.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 50.6%

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

   if 1.0000000000000001e300 < (-.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 75.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   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 56.8%

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

Alternative 9: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(0.6666666666666666, b - c, \sqrt{a} \cdot z\right)}{t}}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{a + t} \cdot \frac{z}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(-\left(b - c\right), 0.8333333333333334 + a, \sqrt{{t}^{-1}} \cdot z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t 4e-110)
   (/
    x
    (+
     x
     (*
      y
      (exp (* 2.0 (/ (fma 0.6666666666666666 (- b c) (* (sqrt a) z)) t))))))
   (if (<= t 3.5e-44)
     (/ x (+ x (* y (exp (* 2.0 (* (sqrt (+ a t)) (/ z t)))))))
     (/
      x
      (+
       x
       (*
        y
        (exp
         (*
          2.0
          (fma
           (- (- b c))
           (+ 0.8333333333333334 a)
           (* (sqrt (pow t -1.0)) z))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= 4e-110) {
		tmp = x / (x + (y * exp((2.0 * (fma(0.6666666666666666, (b - c), (sqrt(a) * z)) / t)))));
	} else if (t <= 3.5e-44) {
		tmp = x / (x + (y * exp((2.0 * (sqrt((a + t)) * (z / t))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * fma(-(b - c), (0.8333333333333334 + a), (sqrt(pow(t, -1.0)) * z))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= 4e-110)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(0.6666666666666666, Float64(b - c), Float64(sqrt(a) * z)) / t))))));
	elseif (t <= 3.5e-44)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(sqrt(Float64(a + t)) * Float64(z / t)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * fma(Float64(-Float64(b - c)), Float64(0.8333333333333334 + a), Float64(sqrt((t ^ -1.0)) * z)))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, 4e-110], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(0.6666666666666666 * N[(b - c), $MachinePrecision] + N[(N[Sqrt[a], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-44], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[((-N[(b - c), $MachinePrecision]) * N[(0.8333333333333334 + a), $MachinePrecision] + N[(N[Sqrt[N[Power[t, -1.0], $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.0000000000000002e-110

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 4.0000000000000002e-110 < t < 3.4999999999999998e-44

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 78.3

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

   8. Applied rewrites 78.3%

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

   if 3.4999999999999998e-44 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 98.3

          \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(-\left(b - c\right), 0.8333333333333334 + a, \sqrt{\color{blue}{\frac{1}{t}}} \cdot z\right)}} \]

   5. Applied rewrites 98.3%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(-\left(b - c\right), 0.8333333333333334 + a, \sqrt{\frac{1}{t}} \cdot z\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(0.6666666666666666, b - c, \sqrt{a} \cdot z\right)}{t}}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\sqrt{a + t} \cdot \frac{z}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(-\left(b - c\right), 0.8333333333333334 + a, \sqrt{{t}^{-1}} \cdot z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \leq -1 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(0.8333333333333334 \cdot c\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))
      -1e+24)
   1.0
   (/ x (+ x (* y (exp (* 2.0 (* 0.8333333333333334 c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))) <= -1e+24) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * exp((2.0 * (0.8333333333333334 * c)))));
	}
	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 ((((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))) <= (-1d+24)) then
        tmp = 1.0d0
    else
        tmp = x / (x + (y * exp((2.0d0 * (0.8333333333333334d0 * c)))))
    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 ((((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))) <= -1e+24) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (0.8333333333333334 * c)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))) <= -1e+24:
		tmp = 1.0
	else:
		tmp = x / (x + (y * math.exp((2.0 * (0.8333333333333334 * c)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (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))))) <= -1e+24)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(0.8333333333333334 * c))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))) <= -1e+24)
		tmp = 1.0;
	else
		tmp = x / (x + (y * exp((2.0 * (0.8333333333333334 * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[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], -1e+24], 1.0, N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(0.8333333333333334 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)))))) < -9.9999999999999998e23

   1. Initial program 98.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.3%

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

   if -9.9999999999999998e23 < (-.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 90.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 49.5%

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(0.8333333333333334 \cdot c\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.0% 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 94.2%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 67.3

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

5. Applied rewrites 67.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 49.9%

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

Developer Target 1: 95.1% 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 2024318 
(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)))))))))))