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

Percentage Accurate: 94.2% → 96.7%

Time: 18.7s

Alternatives: 15

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: 96.7% accurate, 0.7× speedup

Math

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

FPCore

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

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = x / (x + (y * exp((2.0 * t_1))));
	} else {
		tmp = x / fma(y, exp((-2.0 * (b * (a + 0.8333333333333334)))), x);
	}
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * t_1)))));
	else
		tmp = Float64(x / fma(y, exp(Float64(-2.0 * Float64(b * Float64(a + 0.8333333333333334)))), x));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $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[(y * N[Exp[N[(-2.0 * N[(b * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 90.3

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

   8. Simplified 90.3%

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

4. Final simplification 99.6%

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

Alternative 2: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\\ t_2 := \sqrt{t + a}\\ t_3 := \frac{z \cdot t\_2}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 1000000:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2, t\_2 \cdot \frac{z \cdot y}{t}, y\right)}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{x \cdot 0.5}{\left(t\_1 \cdot t\_1\right) \cdot \left(y \cdot \left(c \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(c \cdot \left(\mathsf{fma}\left(2, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), \frac{1}{c}\right) + \frac{x}{c \cdot y}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (- (/ 0.6666666666666666 t) a) 0.8333333333333334))
        (t_2 (sqrt (+ t a)))
        (t_3
         (+
          (/ (* z t_2) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_3 -2e+21)
     1.0
     (if (<= t_3 1000000.0)
       (/ x (+ x (fma 2.0 (* t_2 (/ (* z y) t)) y)))
       (if (<= t_3 4e+133)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (if (<= t_3 2e+283)
           (/ (* x 0.5) (* (* t_1 t_1) (* y (* c c))))
           (/
            x
            (*
             y
             (*
              c
              (+
               (fma
                2.0
                (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t)))
                (/ 1.0 c))
               (/ x (* c y))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((0.6666666666666666 / t) - a) - 0.8333333333333334;
	double t_2 = sqrt((t + a));
	double t_3 = ((z * t_2) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_3 <= -2e+21) {
		tmp = 1.0;
	} else if (t_3 <= 1000000.0) {
		tmp = x / (x + fma(2.0, (t_2 * ((z * y) / t)), y));
	} else if (t_3 <= 4e+133) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else if (t_3 <= 2e+283) {
		tmp = (x * 0.5) / ((t_1 * t_1) * (y * (c * c)));
	} else {
		tmp = x / (y * (c * (fma(2.0, (0.8333333333333334 + (a - (0.6666666666666666 / t))), (1.0 / c)) + (x / (c * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(0.6666666666666666 / t) - a) - 0.8333333333333334)
	t_2 = sqrt(Float64(t + a))
	t_3 = Float64(Float64(Float64(z * t_2) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_3 <= -2e+21)
		tmp = 1.0;
	elseif (t_3 <= 1000000.0)
		tmp = Float64(x / Float64(x + fma(2.0, Float64(t_2 * Float64(Float64(z * y) / t)), y)));
	elseif (t_3 <= 4e+133)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	elseif (t_3 <= 2e+283)
		tmp = Float64(Float64(x * 0.5) / Float64(Float64(t_1 * t_1) * Float64(y * Float64(c * c))));
	else
		tmp = Float64(x / Float64(y * Float64(c * Float64(fma(2.0, Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t))), Float64(1.0 / c)) + Float64(x / Float64(c * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - a), $MachinePrecision] - 0.8333333333333334), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z * t$95$2), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+21], 1.0, If[LessEqual[t$95$3, 1000000.0], N[(x / N[(x + N[(2.0 * N[(t$95$2 * N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+133], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+283], N[(N[(x * 0.5), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(y * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(c * N[(N[(2.0 * N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / c), $MachinePrecision]), $MachinePrecision] + N[(x / N[(c * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 1e6

   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 z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 95.8

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

   8. Simplified 95.8%

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

   if 1e6 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 4.0000000000000001e133

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 56.0

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

   5. Simplified 56.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 20.1

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 20.1%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y \cdot y - x \cdot x}{y - x}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x}} \cdot \left(y - x\right) \]

      5. difference-of-squares N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      11. lower--.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y - x\right)}} \cdot \left(y - x\right) \]

      12. lower--.f64 71.8

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \color{blue}{\left(y - x\right)} \]

   10. Applied egg-rr 71.8%

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \left(y - x\right)} \]

   if 4.0000000000000001e133 < (-.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)))))) < 1.99999999999999991e283

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 67.1

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

   5. Simplified 67.1%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 81.4%

      \[\leadsto \frac{x}{x + y \cdot \color{blue}{\mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}} \]

   9. Taylor expanded in c around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. associate--l+ N/A

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

       12. lower-+.f64 N/A

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

       13. lower--.f64 N/A

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

       14. associate-*r/ N/A

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

       15. metadata-eval N/A

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

       16. lower-/.f64 N/A

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

       17. associate--l+ N/A

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

       18. lower-+.f64 N/A

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

       19. lower--.f64 N/A

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

       20. associate-*r/ N/A

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

       21. metadata-eval N/A

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

       22. lower-/.f64 69.7

           \[\leadsto \frac{0.5 \cdot x}{\left(\left(c \cdot c\right) \cdot y\right) \cdot \left(\left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)\right)} \]

   11. Simplified 69.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\left(\left(c \cdot c\right) \cdot y\right) \cdot \left(\left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right)}} \]

   if 1.99999999999999991e283 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 83.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 64.1

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

   5. Simplified 64.1%

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

   6. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Simplified 56.1%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. associate-+r+ N/A

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

       3. lower-+.f64 N/A

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

       4. +-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. associate--l+ N/A

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

       9. lower-+.f64 N/A

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

       10. lower--.f64 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 62.5

           \[\leadsto \frac{x}{y \cdot \left(\mathsf{fma}\left(2 \cdot c, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right) + \color{blue}{\frac{x}{y}}\right)} \]

   11. Simplified 62.5%

       \[\leadsto \frac{x}{\color{blue}{y \cdot \left(\mathsf{fma}\left(2 \cdot c, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right) + \frac{x}{y}\right)}} \]

   12. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. associate-+r+ N/A

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

       3. lower-+.f64 N/A

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

       4. lower-fma.f64 N/A

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

       5. associate--l+ N/A

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

       6. lower-+.f64 N/A

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

       7. lower--.f64 N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 75.7

           \[\leadsto \frac{x}{y \cdot \left(c \cdot \left(\mathsf{fma}\left(2, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), \frac{1}{c}\right) + \frac{x}{\color{blue}{c \cdot y}}\right)\right)} \]

   14. Simplified 75.7%

       \[\leadsto \frac{x}{y \cdot \color{blue}{\left(c \cdot \left(\mathsf{fma}\left(2, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), \frac{1}{c}\right) + \frac{x}{c \cdot y}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 1000000:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2, \sqrt{t + a} \cdot \frac{z \cdot y}{t}, y\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{x \cdot 0.5}{\left(\left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right) \cdot \left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right)\right) \cdot \left(y \cdot \left(c \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(c \cdot \left(\mathsf{fma}\left(2, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), \frac{1}{c}\right) + \frac{x}{c \cdot y}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 0.8888888888888888 \cdot \frac{c}{t \cdot t}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 2e-20)
       (/ x (+ x y))
       (if (<= t_1 4e+133)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (/
          x
          (+ x (* y (fma c (* 0.8888888888888888 (/ c (* t t))) 1.0)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 2e-20) {
		tmp = x / (x + y);
	} else if (t_1 <= 4e+133) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (x + (y * fma(c, (0.8888888888888888 * (c / (t * t))), 1.0)));
	}
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 2e-20)
		tmp = Float64(x / Float64(x + y));
	elseif (t_1 <= 4e+133)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(0.8888888888888888 * Float64(c / Float64(t * t))), 1.0))));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 2e-20], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+133], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(c * N[(0.8888888888888888 * N[(c / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 1.99999999999999989e-20

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 100.0

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 1.99999999999999989e-20 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 4.0000000000000001e133

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 55.6

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

   5. Simplified 55.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 22.5

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 22.5%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y \cdot y - x \cdot x}{y - x}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x}} \cdot \left(y - x\right) \]

      5. difference-of-squares N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      11. lower--.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y - x\right)}} \cdot \left(y - x\right) \]

      12. lower--.f64 70.3

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \color{blue}{\left(y - x\right)} \]

   10. Applied egg-rr 70.3%

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \left(y - x\right)} \]

   if 4.0000000000000001e133 < (-.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 90.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.3

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

   5. Simplified 65.3%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 83.7%

      \[\leadsto \frac{x}{x + y \cdot \color{blue}{\mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot \mathsf{fma}\left(c, \color{blue}{\frac{8}{9} \cdot \frac{c}{{t}^{2}}}, 1\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{x}{x + y \cdot \mathsf{fma}\left(c, \color{blue}{\frac{8}{9} \cdot \frac{c}{{t}^{2}}}, 1\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{x}{x + y \cdot \mathsf{fma}\left(c, \frac{8}{9} \cdot \color{blue}{\frac{c}{{t}^{2}}}, 1\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{x + y \cdot \mathsf{fma}\left(c, \frac{8}{9} \cdot \frac{c}{\color{blue}{t \cdot t}}, 1\right)} \]

       4. lower-*.f64 62.5

          \[\leadsto \frac{x}{x + y \cdot \mathsf{fma}\left(c, 0.8888888888888888 \cdot \frac{c}{\color{blue}{t \cdot t}}, 1\right)} \]

   11. Simplified 62.5%

       \[\leadsto \frac{x}{x + y \cdot \mathsf{fma}\left(c, \color{blue}{0.8888888888888888 \cdot \frac{c}{t \cdot t}}, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 0.8888888888888888 \cdot \frac{c}{t \cdot t}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(c \cdot 2, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 2e-20)
       (/ x (+ x y))
       (if (<= t_1 5e+282)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (/
          x
          (*
           y
           (fma
            (* c 2.0)
            (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t)))
            1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 2e-20) {
		tmp = x / (x + y);
	} else if (t_1 <= 5e+282) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (y * fma((c * 2.0), (0.8333333333333334 + (a - (0.6666666666666666 / t))), 1.0));
	}
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 2e-20)
		tmp = Float64(x / Float64(x + y));
	elseif (t_1 <= 5e+282)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = Float64(x / Float64(y * fma(Float64(c * 2.0), Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t))), 1.0)));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 2e-20], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+282], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(N[(c * 2.0), $MachinePrecision] * N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 1.99999999999999989e-20

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 100.0

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 1.99999999999999989e-20 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 4.99999999999999978e282

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 17.0

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 17.0%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y \cdot y - x \cdot x}{y - x}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x}} \cdot \left(y - x\right) \]

      5. difference-of-squares N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      11. lower--.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y - x\right)}} \cdot \left(y - x\right) \]

      12. lower--.f64 53.9

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \color{blue}{\left(y - x\right)} \]

   10. Applied egg-rr 53.9%

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \left(y - x\right)} \]

   if 4.99999999999999978e282 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 83.6%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Simplified 54.5%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate--l+ N/A

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

       7. lower-+.f64 N/A

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

       8. lower--.f64 N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f64 62.3

           \[\leadsto \frac{x}{y \cdot \mathsf{fma}\left(2 \cdot c, 0.8333333333333334 + \left(a - \color{blue}{\frac{0.6666666666666666}{t}}\right), 1\right)} \]

   11. Simplified 62.3%

       \[\leadsto \frac{x}{\color{blue}{y \cdot \mathsf{fma}\left(2 \cdot c, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(c \cdot 2, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.8888888888888888 \cdot \frac{y \cdot \left(c \cdot c\right)}{t \cdot t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 2e-20)
       (/ x (+ x y))
       (if (<= t_1 5e+262)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (/ x (* 0.8888888888888888 (/ (* y (* c c)) (* t 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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 2e-20) {
		tmp = x / (x + y);
	} else if (t_1 <= 5e+262) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (0.8888888888888888 * ((y * (c * c)) / (t * t)));
	}
	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) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))
    if (t_1 <= (-2d+21)) then
        tmp = 1.0d0
    else if (t_1 <= 2d-20) then
        tmp = x / (x + y)
    else if (t_1 <= 5d+262) then
        tmp = (y - x) * (x / ((x + y) * (y - x)))
    else
        tmp = x / (0.8888888888888888d0 * ((y * (c * c)) / (t * t)))
    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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 2e-20) {
		tmp = x / (x + y);
	} else if (t_1 <= 5e+262) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (0.8888888888888888 * ((y * (c * c)) / (t * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))
	tmp = 0
	if t_1 <= -2e+21:
		tmp = 1.0
	elif t_1 <= 2e-20:
		tmp = x / (x + y)
	elif t_1 <= 5e+262:
		tmp = (y - x) * (x / ((x + y) * (y - x)))
	else:
		tmp = x / (0.8888888888888888 * ((y * (c * c)) / (t * 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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 2e-20)
		tmp = Float64(x / Float64(x + y));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = Float64(x / Float64(0.8888888888888888 * Float64(Float64(y * Float64(c * c)) / Float64(t * t))));
	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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	tmp = 0.0;
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 2e-20)
		tmp = x / (x + y);
	elseif (t_1 <= 5e+262)
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	else
		tmp = x / (0.8888888888888888 * ((y * (c * c)) / (t * t)));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 2e-20], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(0.8888888888888888 * N[(N[(y * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(t * t), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 1.99999999999999989e-20

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 100.0

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 100.0%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 1.99999999999999989e-20 < (-.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)))))) < 5.00000000000000008e262

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 49.4

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

   5. Simplified 49.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 17.9

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 17.9%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y \cdot y - x \cdot x}{y - x}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x}} \cdot \left(y - x\right) \]

      5. difference-of-squares N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      11. lower--.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y - x\right)}} \cdot \left(y - x\right) \]

      12. lower--.f64 55.6

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \color{blue}{\left(y - x\right)} \]

   10. Applied egg-rr 55.6%

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \left(y - x\right)} \]

   if 5.00000000000000008e262 < (-.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 84.6%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 86.6%

      \[\leadsto \frac{x}{x + y \cdot \color{blue}{\mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x}{\color{blue}{\frac{8}{9} \cdot \frac{{c}^{2} \cdot y}{{t}^{2}}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\frac{8}{9} \cdot \frac{{c}^{2} \cdot y}{{t}^{2}}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{x}{\frac{8}{9} \cdot \color{blue}{\frac{{c}^{2} \cdot y}{{t}^{2}}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{x}{\frac{8}{9} \cdot \frac{\color{blue}{{c}^{2} \cdot y}}{{t}^{2}}} \]

       4. unpow2 N/A

          \[\leadsto \frac{x}{\frac{8}{9} \cdot \frac{\color{blue}{\left(c \cdot c\right)} \cdot y}{{t}^{2}}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{x}{\frac{8}{9} \cdot \frac{\color{blue}{\left(c \cdot c\right)} \cdot y}{{t}^{2}}} \]

       6. unpow2 N/A

          \[\leadsto \frac{x}{\frac{8}{9} \cdot \frac{\left(c \cdot c\right) \cdot y}{\color{blue}{t \cdot t}}} \]

       7. lower-*.f64 59.1

          \[\leadsto \frac{x}{0.8888888888888888 \cdot \frac{\left(c \cdot c\right) \cdot y}{\color{blue}{t \cdot t}}} \]

   11. Simplified 59.1%

       \[\leadsto \frac{x}{\color{blue}{0.8888888888888888 \cdot \frac{\left(c \cdot c\right) \cdot y}{t \cdot t}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{0.8888888888888888 \cdot \frac{y \cdot \left(c \cdot c\right)}{t \cdot t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
      5e-45)
   (* (- y x) (/ x (* (+ x y) (- y x))))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0)))))))))) <= 5d-45) then
        tmp = (y - x) * (x / ((x + y) * (y - x)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45:
		tmp = (y - x) * (x / ((x + y) * (y - x)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))))))) <= 5e-45)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45)
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-45], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999976e-45

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 22.3

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 22.3%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y \cdot y - x \cdot x}{y - x}}} \]

      2. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x} \cdot \left(y - x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y - x \cdot x}} \cdot \left(y - x\right) \]

      5. difference-of-squares N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y - x\right)}} \cdot \left(y - x\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(y - x\right)} \cdot \left(y - x\right) \]

      11. lower--.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y - x\right)}} \cdot \left(y - x\right) \]

      12. lower--.f64 53.1

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \color{blue}{\left(y - x\right)} \]

   10. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(y - x\right)} \cdot \left(y - x\right)} \]

   if 4.99999999999999976e-45 < (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))))))

   1. Initial program 92.4%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 71.1

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

   5. Simplified 71.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 96.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
      5e-45)
   (/ x (+ x y))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45) {
		tmp = x / (x + y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0)))))))))) <= 5d-45) then
        tmp = x / (x + y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45) {
		tmp = x / (x + y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45:
		tmp = x / (x + y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))))))) <= 5e-45)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45)
		tmp = x / (x + y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-45], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999976e-45

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 22.3

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 22.3%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 4.99999999999999976e-45 < (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))))))

   1. Initial program 92.4%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 71.1

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

   5. Simplified 71.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 96.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
      5e-45)
   (/ x y)
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0)))))))))) <= 5d-45) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))))))) <= 5e-45)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 5e-45)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-45], N[(x / y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999976e-45

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 22.3

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 22.3%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. lower-/.f64 21.9

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Simplified 21.9%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 4.99999999999999976e-45 < (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))))))

   1. Initial program 92.4%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 71.1

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

   5. Simplified 71.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 96.3%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{c \cdot \left(y \cdot \left(a + 0.8333333333333334\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 2e+24)
       (/ x (+ x y))
       (/ (* x 0.5) (* c (* y (+ a 0.8333333333333334))))))))

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 2e+24) {
		tmp = x / (x + y);
	} else {
		tmp = (x * 0.5) / (c * (y * (a + 0.8333333333333334)));
	}
	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) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))
    if (t_1 <= (-2d+21)) then
        tmp = 1.0d0
    else if (t_1 <= 2d+24) then
        tmp = x / (x + y)
    else
        tmp = (x * 0.5d0) / (c * (y * (a + 0.8333333333333334d0)))
    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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 2e+24) {
		tmp = x / (x + y);
	} else {
		tmp = (x * 0.5) / (c * (y * (a + 0.8333333333333334)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))
	tmp = 0
	if t_1 <= -2e+21:
		tmp = 1.0
	elif t_1 <= 2e+24:
		tmp = x / (x + y)
	else:
		tmp = (x * 0.5) / (c * (y * (a + 0.8333333333333334)))
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 2e+24)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(Float64(x * 0.5) / Float64(c * Float64(y * Float64(a + 0.8333333333333334))));
	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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	tmp = 0.0;
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 2e+24)
		tmp = x / (x + y);
	else
		tmp = (x * 0.5) / (c * (y * (a + 0.8333333333333334)));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 2e+24], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(c * N[(y * N[(a + 0.8333333333333334), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 2e24

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 92.5

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

   5. Simplified 92.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 88.8

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 88.8%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 2e24 < (-.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 91.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Simplified 40.6%

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

   9. Taylor expanded in c around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. associate--l+ N/A

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

       8. lower-+.f64 N/A

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

       9. lower--.f64 N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 38.6

           \[\leadsto \frac{0.5 \cdot x}{\left(c \cdot y\right) \cdot \left(0.8333333333333334 + \left(a - \color{blue}{\frac{0.6666666666666666}{t}}\right)\right)} \]

   11. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\left(c \cdot y\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)}} \]

   12. Taylor expanded in t around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-+.f64 36.2

          \[\leadsto \frac{0.5 \cdot x}{c \cdot \left(y \cdot \color{blue}{\left(0.8333333333333334 + a\right)}\right)} \]

   14. Simplified 36.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot x}{c \cdot \left(y \cdot \left(0.8333333333333334 + a\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{c \cdot \left(y \cdot \left(a + 0.8333333333333334\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(a \cdot \left(c \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 5e+137) (/ x (+ x y)) (/ x (+ x (* 2.0 (* a (* c y)))))))))

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 5e+137) {
		tmp = x / (x + y);
	} else {
		tmp = x / (x + (2.0 * (a * (c * y))));
	}
	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) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))
    if (t_1 <= (-2d+21)) then
        tmp = 1.0d0
    else if (t_1 <= 5d+137) then
        tmp = x / (x + y)
    else
        tmp = x / (x + (2.0d0 * (a * (c * y))))
    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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 5e+137) {
		tmp = x / (x + y);
	} else {
		tmp = x / (x + (2.0 * (a * (c * y))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))
	tmp = 0
	if t_1 <= -2e+21:
		tmp = 1.0
	elif t_1 <= 5e+137:
		tmp = x / (x + y)
	else:
		tmp = x / (x + (2.0 * (a * (c * y))))
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 5e+137)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(x / Float64(x + Float64(2.0 * Float64(a * Float64(c * y)))));
	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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	tmp = 0.0;
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 5e+137)
		tmp = x / (x + y);
	else
		tmp = x / (x + (2.0 * (a * (c * y))));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 5e+137], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(2.0 * N[(a * N[(c * y), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 5.0000000000000002e137

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 72.5

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

   5. Simplified 72.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 54.9

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 54.9%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 5.0000000000000002e137 < (-.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 89.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.6

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

   5. Simplified 65.6%

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

   6. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Simplified 44.3%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 31.6

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

   11. Simplified 31.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(a \cdot \left(c \cdot y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.75 \cdot \left(t \cdot x\right)}{c \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 4e+133) (/ x (+ x y)) (/ (* -0.75 (* t x)) (* c y))))))

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 4e+133) {
		tmp = x / (x + y);
	} else {
		tmp = (-0.75 * (t * x)) / (c * y);
	}
	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) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))
    if (t_1 <= (-2d+21)) then
        tmp = 1.0d0
    else if (t_1 <= 4d+133) then
        tmp = x / (x + y)
    else
        tmp = ((-0.75d0) * (t * x)) / (c * y)
    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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 4e+133) {
		tmp = x / (x + y);
	} else {
		tmp = (-0.75 * (t * x)) / (c * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))
	tmp = 0
	if t_1 <= -2e+21:
		tmp = 1.0
	elif t_1 <= 4e+133:
		tmp = x / (x + y)
	else:
		tmp = (-0.75 * (t * x)) / (c * y)
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 4e+133)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(Float64(-0.75 * Float64(t * x)) / Float64(c * y));
	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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	tmp = 0.0;
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 4e+133)
		tmp = x / (x + y);
	else
		tmp = (-0.75 * (t * x)) / (c * y);
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 4e+133], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(-0.75 * N[(t * x), $MachinePrecision]), $MachinePrecision] / N[(c * y), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 4.0000000000000001e133

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 57.1

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 57.1%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 4.0000000000000001e133 < (-.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 90.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.3

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

   5. Simplified 65.3%

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

   6. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Simplified 43.5%

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

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{-3}{4} \cdot \frac{t \cdot x}{c \cdot y}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-3}{4} \cdot \left(t \cdot x\right)}{c \cdot y}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-3}{4} \cdot \left(t \cdot x\right)}{c \cdot y}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-3}{4} \cdot \left(t \cdot x\right)}}{c \cdot y} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-3}{4} \cdot \color{blue}{\left(t \cdot x\right)}}{c \cdot y} \]

       5. lower-*.f64 30.4

          \[\leadsto \frac{-0.75 \cdot \left(t \cdot x\right)}{\color{blue}{c \cdot y}} \]

   11. Simplified 30.4%

       \[\leadsto \color{blue}{\frac{-0.75 \cdot \left(t \cdot x\right)}{c \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.75 \cdot \left(t \cdot x\right)}{c \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+235}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(2 \cdot \left(a \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+21)
     1.0
     (if (<= t_1 1e+235) (/ x (+ x y)) (/ x (* y (* 2.0 (* a c))))))))

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 1e+235) {
		tmp = x / (x + y);
	} else {
		tmp = x / (y * (2.0 * (a * 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) :: t_1
    real(8) :: tmp
    t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))
    if (t_1 <= (-2d+21)) then
        tmp = 1.0d0
    else if (t_1 <= 1d+235) then
        tmp = x / (x + y)
    else
        tmp = x / (y * (2.0d0 * (a * 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 t_1 = ((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+21) {
		tmp = 1.0;
	} else if (t_1 <= 1e+235) {
		tmp = x / (x + y);
	} else {
		tmp = x / (y * (2.0 * (a * c)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))
	tmp = 0
	if t_1 <= -2e+21:
		tmp = 1.0
	elif t_1 <= 1e+235:
		tmp = x / (x + y)
	else:
		tmp = x / (y * (2.0 * (a * c)))
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 1e+235)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(x / Float64(y * Float64(2.0 * Float64(a * c))));
	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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	tmp = 0.0;
	if (t_1 <= -2e+21)
		tmp = 1.0;
	elseif (t_1 <= 1e+235)
		tmp = x / (x + y);
	else
		tmp = x / (y * (2.0 * (a * c)));
	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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+21], 1.0, If[LessEqual[t$95$1, 1e+235], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(2.0 * N[(a * c), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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)))))) < 1.0000000000000001e235

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 64.8

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

   5. Simplified 64.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{x + y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

      2. lower-+.f64 42.7

         \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   8. Simplified 42.7%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]

   if 1.0000000000000001e235 < (-.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 86.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Simplified 51.4%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. associate-+r+ N/A

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

       3. lower-+.f64 N/A

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

       4. +-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. associate--l+ N/A

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

       9. lower-+.f64 N/A

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

       10. lower--.f64 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f64 N/A

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

       14. lower-/.f64 57.9

           \[\leadsto \frac{x}{y \cdot \left(\mathsf{fma}\left(2 \cdot c, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right) + \color{blue}{\frac{x}{y}}\right)} \]

   11. Simplified 57.9%

       \[\leadsto \frac{x}{\color{blue}{y \cdot \left(\mathsf{fma}\left(2 \cdot c, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right), 1\right) + \frac{x}{y}\right)}} \]

   12. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 32.9

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

   14. Simplified 32.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+235}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(2 \cdot \left(a \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\\ \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, t\_1 \cdot t\_1, 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (- (/ 0.6666666666666666 t) a) 0.8333333333333334)))
   (if (<=
        (+
         (/ (* z (sqrt (+ t a))) t)
         (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))
        -2e+21)
     1.0
     (/
      x
      (+
       x
       (*
        y
        (fma
         c
         (*
          2.0
          (fma
           c
           (* t_1 t_1)
           (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t)))))
         1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((0.6666666666666666 / t) - a) - 0.8333333333333334;
	double tmp;
	if ((((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -2e+21) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * fma(c, (t_1 * t_1), (0.8333333333333334 + (a - (0.6666666666666666 / t))))), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(0.6666666666666666 / t) - a) - 0.8333333333333334)
	tmp = 0.0
	if (Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0))))) <= -2e+21)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(2.0 * fma(c, Float64(t_1 * t_1), Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t))))), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - a), $MachinePrecision] - 0.8333333333333334), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+21], 1.0, N[(x / N[(x + N[(y * N[(c * N[(2.0 * N[(c * N[(t$95$1 * t$95$1), $MachinePrecision] + N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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 93.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Simplified 79.7%

      \[\leadsto \frac{x}{x + y \cdot \color{blue}{\mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right) \cdot \left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (+
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))
      -2e+21)
   1.0
   (/
    x
    (fma
     y
     (fma
      c
      (*
       2.0
       (fma
        c
        (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))
        (+ a 0.8333333333333334)))
      1.0)
     x))))

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -2e+21) {
		tmp = 1.0;
	} else {
		tmp = x / fma(y, fma(c, (2.0 * fma(c, ((a + 0.8333333333333334) * (a + 0.8333333333333334)), (a + 0.8333333333333334))), 1.0), x);
	}
	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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0))))) <= -2e+21)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(y, fma(c, Float64(2.0 * fma(c, Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334)), Float64(a + 0.8333333333333334))), 1.0), x));
	end
	return 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[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+21], 1.0, N[(x / N[(y * N[(c * N[(2.0 * N[(c * N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e21

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -2e21 < (-.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 93.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 55.4

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

   8. Simplified 55.4%

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

   9. Taylor expanded in c around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(2 \cdot \left(c \cdot {\left(\frac{5}{6} + a\right)}^{2}\right) + 2 \cdot \left(\frac{5}{6} + a\right)\right)}, x\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. distribute-lft-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \color{blue}{\left(\frac{5}{6} + a\right) \cdot \left(\frac{5}{6} + a\right)}, \frac{5}{6} + a\right), 1\right), x\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \color{blue}{\left(\frac{5}{6} + a\right) \cdot \left(\frac{5}{6} + a\right)}, \frac{5}{6} + a\right), 1\right), x\right)} \]

       8. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \color{blue}{\left(a + \frac{5}{6}\right)} \cdot \left(\frac{5}{6} + a\right), \frac{5}{6} + a\right), 1\right), x\right)} \]

       9. lower-+.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \color{blue}{\left(a + \frac{5}{6}\right)} \cdot \left(\frac{5}{6} + a\right), \frac{5}{6} + a\right), 1\right), x\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + \frac{5}{6}\right) \cdot \color{blue}{\left(a + \frac{5}{6}\right)}, \frac{5}{6} + a\right), 1\right), x\right)} \]

       11. lower-+.f64 N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + \frac{5}{6}\right) \cdot \color{blue}{\left(a + \frac{5}{6}\right)}, \frac{5}{6} + a\right), 1\right), x\right)} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + \frac{5}{6}\right) \cdot \left(a + \frac{5}{6}\right), \color{blue}{a + \frac{5}{6}}\right), 1\right), x\right)} \]

       13. lower-+.f64 67.9

           \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), \color{blue}{a + 0.8333333333333334}\right), 1\right), x\right)} \]

   11. Simplified 67.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.2% accurate, 198.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 65.5

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

5. Simplified 65.5%

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

6. Taylor expanded in x around inf

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

8. Simplified 51.2%

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

Developer Target 1: 95.2% 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 2024207 
(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)))))))))))