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

Percentage Accurate: 93.8% → 93.8%

Time: 21.2s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% 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: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))));
}

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) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0)))))))))
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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))));
}

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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))

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(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0))))))))))
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) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))));
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[(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]

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 2: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{t + a}\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot t\_1}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{y \cdot \left(e^{2 \cdot \mathsf{fma}\left(\frac{z}{t}, t\_1, \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(c - b\right)\right)} + \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (sqrt (+ t a))))
   (if (<=
        (/
         x
         (+
          x
          (*
           y
           (exp
            (*
             2.0
             (+
              (/ (* z t_1) t)
              (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
        0.9999999999999916)
     (/
      x
      (*
       y
       (+
        (exp
         (*
          2.0
          (fma
           (/ z t)
           t_1
           (*
            (+ a (+ 0.8333333333333334 (/ -0.6666666666666666 t)))
            (- c b)))))
        (/ x y))))
     1.0)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = sqrt((t + a));
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * t_1) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 0.9999999999999916) {
		tmp = x / (y * (exp((2.0 * fma((z / t), t_1, ((a + (0.8333333333333334 + (-0.6666666666666666 / t))) * (c - b))))) + (x / y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = sqrt(Float64(t + a))
	tmp = 0.0
	if (Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * t_1) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))))))) <= 0.9999999999999916)
		tmp = Float64(x / Float64(y * Float64(exp(Float64(2.0 * fma(Float64(z / t), t_1, Float64(Float64(a + Float64(0.8333333333333334 + Float64(-0.6666666666666666 / t))) * Float64(c - b))))) + Float64(x / y))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * t$95$1), $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], 0.9999999999999916], N[(x / N[(y * N[(N[Exp[N[(2.0 * N[(N[(z / t), $MachinePrecision] * t$95$1 + N[(N[(a + N[(0.8333333333333334 + N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(x / y), $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))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 91.0%

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

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{y \cdot \left(e^{2 \cdot \mathsf{fma}\left(\frac{z}{t}, \sqrt{t + a}, \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right) \cdot \left(c - b\right)\right)} + \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ t_2 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ t_3 := \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+126}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot t\_3\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(t\_2 \cdot t\_2\right), y \cdot t\_2\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, t\_3 \cdot t\_3, t\_3\right), 1\right)}\\ \end{array} \end{array} \]

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))))))
        (t_2 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t))))
        (t_3 (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334))))
   (if (<= t_1 -5e+126)
     1.0
     (if (<= t_1 2e+177)
       (/ x (+ x (* y (exp (* 2.0 (* b t_3))))))
       (if (<= t_1 4e+288)
         (/ x (+ x (fma c (* 2.0 (fma c (* y (* t_2 t_2)) (* y t_2))) y)))
         (/ x (+ x (* y (fma b (* 2.0 (fma b (* t_3 t_3) t_3)) 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 t_2 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	double t_3 = (0.6666666666666666 / t) - (a + 0.8333333333333334);
	double tmp;
	if (t_1 <= -5e+126) {
		tmp = 1.0;
	} else if (t_1 <= 2e+177) {
		tmp = x / (x + (y * exp((2.0 * (b * t_3)))));
	} else if (t_1 <= 4e+288) {
		tmp = x / (x + fma(c, (2.0 * fma(c, (y * (t_2 * t_2)), (y * t_2))), y));
	} else {
		tmp = x / (x + (y * fma(b, (2.0 * fma(b, (t_3 * t_3), t_3)), 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)))))
	t_2 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	t_3 = Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))
	tmp = 0.0
	if (t_1 <= -5e+126)
		tmp = 1.0;
	elseif (t_1 <= 2e+177)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(b * t_3))))));
	elseif (t_1 <= 4e+288)
		tmp = Float64(x / Float64(x + fma(c, Float64(2.0 * fma(c, Float64(y * Float64(t_2 * t_2)), Float64(y * t_2))), y)));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(b, Float64(2.0 * fma(b, Float64(t_3 * t_3), t_3)), 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]}, Block[{t$95$2 = N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+126], 1.0, If[LessEqual[t$95$1, 2e+177], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[(x / N[(x + N[(c * N[(2.0 * N[(c * N[(y * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(b * N[(2.0 * N[(b * N[(t$95$3 * t$95$3), $MachinePrecision] + t$95$3), $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)))))) < -4.99999999999999977e126

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.0%

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

      1. *-inverses 99.0

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

   7. Applied egg-rr 99.0%

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

   if -4.99999999999999977e126 < (-.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)))))) < 2e177

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 76.7

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

   5. Simplified 76.7%

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

   if 2e177 < (-.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)))))) < 4e288

   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 82.0

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

   5. Simplified 82.0%

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

   6. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 79.1%

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

   if 4e288 < (-.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 94.8%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 83.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -5 \cdot 10^{+126}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+177}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right)\right)}}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(\left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\\ t_2 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ t_3 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_3 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \left(c \cdot \left(a + 0.8333333333333334\right)\right)}, x\right)}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(t\_2 \cdot t\_2\right), y \cdot t\_2\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334)))
        (t_2 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t))))
        (t_3
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_3 -200.0)
     1.0
     (if (<= t_3 4e+146)
       (/ x (fma y (exp (* 2.0 (* c (+ a 0.8333333333333334)))) x))
       (if (<= t_3 4e+288)
         (/ x (+ x (fma c (* 2.0 (fma c (* y (* t_2 t_2)) (* y t_2))) y)))
         (/ x (+ x (* y (fma b (* 2.0 (fma b (* t_1 t_1) t_1)) 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 t_2 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	double t_3 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_3 <= -200.0) {
		tmp = 1.0;
	} else if (t_3 <= 4e+146) {
		tmp = x / fma(y, exp((2.0 * (c * (a + 0.8333333333333334)))), x);
	} else if (t_3 <= 4e+288) {
		tmp = x / (x + fma(c, (2.0 * fma(c, (y * (t_2 * t_2)), (y * t_2))), y));
	} else {
		tmp = x / (x + (y * fma(b, (2.0 * fma(b, (t_1 * t_1), t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))
	t_2 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	t_3 = 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_3 <= -200.0)
		tmp = 1.0;
	elseif (t_3 <= 4e+146)
		tmp = Float64(x / fma(y, exp(Float64(2.0 * Float64(c * Float64(a + 0.8333333333333334)))), x));
	elseif (t_3 <= 4e+288)
		tmp = Float64(x / Float64(x + fma(c, Float64(2.0 * fma(c, Float64(y * Float64(t_2 * t_2)), Float64(y * t_2))), y)));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(b, Float64(2.0 * fma(b, Float64(t_1 * t_1), t_1)), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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$3, -200.0], 1.0, If[LessEqual[t$95$3, 4e+146], N[(x / N[(y * N[Exp[N[(2.0 * N[(c * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+288], N[(x / N[(x + N[(c * N[(2.0 * N[(c * N[(y * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(b * N[(2.0 * N[(b * N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$1), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 3.99999999999999973e146

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 75.1

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

   5. Simplified 75.1%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 70.8

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

   8. Simplified 70.8%

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

   if 3.99999999999999973e146 < (-.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)))))) < 4e288

   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 75.6%

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

   if 4e288 < (-.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 94.8%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 83.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \left(c \cdot \left(a + 0.8333333333333334\right)\right)}, x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(\left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\\ t_2 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ t_3 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_3 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{-2 \cdot \left(b \cdot \left(a + 0.8333333333333334\right)\right)}, x\right)}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(t\_2 \cdot t\_2\right), y \cdot t\_2\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334)))
        (t_2 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t))))
        (t_3
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_3 -200.0)
     1.0
     (if (<= t_3 2e+138)
       (/ x (fma y (exp (* -2.0 (* b (+ a 0.8333333333333334)))) x))
       (if (<= t_3 4e+288)
         (/ x (+ x (fma c (* 2.0 (fma c (* y (* t_2 t_2)) (* y t_2))) y)))
         (/ x (+ x (* y (fma b (* 2.0 (fma b (* t_1 t_1) t_1)) 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 t_2 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	double t_3 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_3 <= -200.0) {
		tmp = 1.0;
	} else if (t_3 <= 2e+138) {
		tmp = x / fma(y, exp((-2.0 * (b * (a + 0.8333333333333334)))), x);
	} else if (t_3 <= 4e+288) {
		tmp = x / (x + fma(c, (2.0 * fma(c, (y * (t_2 * t_2)), (y * t_2))), y));
	} else {
		tmp = x / (x + (y * fma(b, (2.0 * fma(b, (t_1 * t_1), t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))
	t_2 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	t_3 = 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_3 <= -200.0)
		tmp = 1.0;
	elseif (t_3 <= 2e+138)
		tmp = Float64(x / fma(y, exp(Float64(-2.0 * Float64(b * Float64(a + 0.8333333333333334)))), x));
	elseif (t_3 <= 4e+288)
		tmp = Float64(x / Float64(x + fma(c, Float64(2.0 * fma(c, Float64(y * Float64(t_2 * t_2)), Float64(y * t_2))), y)));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(b, Float64(2.0 * fma(b, Float64(t_1 * t_1), t_1)), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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$3, -200.0], 1.0, If[LessEqual[t$95$3, 2e+138], N[(x / N[(y * N[Exp[N[(-2.0 * N[(b * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+288], N[(x / N[(x + N[(c * N[(2.0 * N[(c * N[(y * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(b * N[(2.0 * N[(b * N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$1), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 2.0000000000000001e138

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.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. +-lowering-+.f64 70.3

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

   8. Simplified 70.3%

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

   if 2.0000000000000001e138 < (-.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)))))) < 4e288

   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 73.5%

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

   if 4e288 < (-.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 94.8%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 83.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{-2 \cdot \left(b \cdot \left(a + 0.8333333333333334\right)\right)}, x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(\left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\\ t_2 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ t_3 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_3 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{c \cdot 1.6666666666666667}, x\right)}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(t\_2 \cdot t\_2\right), y \cdot t\_2\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334)))
        (t_2 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t))))
        (t_3
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_3 -200.0)
     1.0
     (if (<= t_3 2e+104)
       (/ x (fma y (exp (* c 1.6666666666666667)) x))
       (if (<= t_3 4e+288)
         (/ x (+ x (fma c (* 2.0 (fma c (* y (* t_2 t_2)) (* y t_2))) y)))
         (/ x (+ x (* y (fma b (* 2.0 (fma b (* t_1 t_1) t_1)) 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 t_2 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	double t_3 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_3 <= -200.0) {
		tmp = 1.0;
	} else if (t_3 <= 2e+104) {
		tmp = x / fma(y, exp((c * 1.6666666666666667)), x);
	} else if (t_3 <= 4e+288) {
		tmp = x / (x + fma(c, (2.0 * fma(c, (y * (t_2 * t_2)), (y * t_2))), y));
	} else {
		tmp = x / (x + (y * fma(b, (2.0 * fma(b, (t_1 * t_1), t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))
	t_2 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	t_3 = 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_3 <= -200.0)
		tmp = 1.0;
	elseif (t_3 <= 2e+104)
		tmp = Float64(x / fma(y, exp(Float64(c * 1.6666666666666667)), x));
	elseif (t_3 <= 4e+288)
		tmp = Float64(x / Float64(x + fma(c, Float64(2.0 * fma(c, Float64(y * Float64(t_2 * t_2)), Float64(y * t_2))), y)));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(b, Float64(2.0 * fma(b, Float64(t_1 * t_1), t_1)), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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$3, -200.0], 1.0, If[LessEqual[t$95$3, 2e+104], N[(x / N[(y * N[Exp[N[(c * 1.6666666666666667), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+288], N[(x / N[(x + N[(c * N[(2.0 * N[(c * N[(y * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(b * N[(2.0 * N[(b * N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$1), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 2e104

   1. Initial program 99.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 83.7

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

   5. Simplified 83.7%

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

   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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 78.1

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

   8. Simplified 78.1%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 78.1

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 78.1%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   if 2e104 < (-.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)))))) < 4e288

   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 68.0

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

   5. Simplified 68.0%

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

   6. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 68.2%

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

   if 4e288 < (-.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 94.8%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 83.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{c \cdot 1.6666666666666667}, x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(\left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\\ t_2 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_2 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334)))
        (t_2
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_2 -200.0)
     1.0
     (if (<= t_2 5e+280)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* c (+ a (+ 0.8333333333333334 (/ -0.6666666666666666 t)))))))))
       (/ x (+ x (* y (fma b (* 2.0 (fma b (* t_1 t_1) t_1)) 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 t_2 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_2 <= -200.0) {
		tmp = 1.0;
	} else if (t_2 <= 5e+280) {
		tmp = x / (x + (y * exp((2.0 * (c * (a + (0.8333333333333334 + (-0.6666666666666666 / t))))))));
	} else {
		tmp = x / (x + (y * fma(b, (2.0 * fma(b, (t_1 * t_1), t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))
	t_2 = 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_2 <= -200.0)
		tmp = 1.0;
	elseif (t_2 <= 5e+280)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * Float64(a + Float64(0.8333333333333334 + Float64(-0.6666666666666666 / t)))))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(b, Float64(2.0 * fma(b, Float64(t_1 * t_1), t_1)), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$2, -200.0], 1.0, If[LessEqual[t$95$2, 5e+280], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * N[(a + N[(0.8333333333333334 + N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(b * N[(2.0 * N[(b * N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 5.0000000000000002e280

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 73.9

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

   5. Simplified 73.9%

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

   if 5.0000000000000002e280 < (-.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 95.0%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.7

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

   5. Simplified 68.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 83.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334))))
   (if (<=
        (/
         x
         (+
          x
          (*
           y
           (exp
            (*
             2.0
             (+
              (/ (* z (sqrt (+ t a))) t)
              (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
        0.9999999999999916)
     (/ x (+ x (* y (fma b (* 2.0 (fma b (* t_1 t_1) t_1)) 1.0))))
     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 ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))))) <= 0.9999999999999916) {
		tmp = x / (x + (y * fma(b, (2.0 * fma(b, (t_1 * t_1), t_1)), 1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))
	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)))))))))) <= 0.9999999999999916)
		tmp = Float64(x / Float64(x + Float64(y * fma(b, Float64(2.0 * fma(b, Float64(t_1 * t_1), t_1)), 1.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, 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], 0.9999999999999916], N[(x / N[(x + N[(y * N[(b * N[(2.0 * N[(b * N[(t$95$1 * t$95$1), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $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))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 63.0

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

   5. Simplified 63.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 70.6%

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

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(b, 2 \cdot \mathsf{fma}\left(b, \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), \frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot 1.3333333333333333, \left(a + 0.8333333333333334\right) \cdot t\_1, 2 \cdot t\_1\right), 1.6666666666666667 + 2 \cdot a\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))))
   (if (<=
        (/
         x
         (+
          x
          (*
           y
           (exp
            (*
             2.0
             (+
              (/ (* z (sqrt (+ t a))) t)
              (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
        0.9999999999999916)
     (/
      x
      (fma
       y
       (fma
        c
        (fma
         c
         (fma
          (* c 1.3333333333333333)
          (* (+ a 0.8333333333333334) t_1)
          (* 2.0 t_1))
         (+ 1.6666666666666667 (* 2.0 a)))
        1.0)
       x))
     1.0)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a + 0.8333333333333334) * (a + 0.8333333333333334);
	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)))))))))) <= 0.9999999999999916) {
		tmp = x / fma(y, fma(c, fma(c, fma((c * 1.3333333333333333), ((a + 0.8333333333333334) * t_1), (2.0 * t_1)), (1.6666666666666667 + (2.0 * a))), 1.0), x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334))
	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)))))))))) <= 0.9999999999999916)
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(Float64(c * 1.3333333333333333), Float64(Float64(a + 0.8333333333333334) * t_1), Float64(2.0 * t_1)), Float64(1.6666666666666667 + Float64(2.0 * a))), 1.0), x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, 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], 0.9999999999999916], N[(x / N[(y * N[(c * N[(c * N[(N[(c * 1.3333333333333333), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.6666666666666667 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(2 \cdot \left(\frac{5}{6} + a\right) + c \cdot \left(\frac{4}{3} \cdot \left(c \cdot {\left(\frac{5}{6} + a\right)}^{3}\right) + 2 \cdot {\left(\frac{5}{6} + a\right)}^{2}\right)\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, 2 \cdot \left(\frac{5}{6} + a\right) + c \cdot \left(\frac{4}{3} \cdot \left(c \cdot {\left(\frac{5}{6} + a\right)}^{3}\right) + 2 \cdot {\left(\frac{5}{6} + a\right)}^{2}\right), 1\right)}, x\right)} \]

   11. Simplified 60.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(1.3333333333333333 \cdot c, \left(0.8333333333333334 + a\right) \cdot \left(\left(0.8333333333333334 + a\right) \cdot \left(0.8333333333333334 + a\right)\right), 2 \cdot \left(\left(0.8333333333333334 + a\right) \cdot \left(0.8333333333333334 + a\right)\right)\right), 1.6666666666666667 + 2 \cdot a\right), 1\right)}, x\right)} \]

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot 1.3333333333333333, \left(a + 0.8333333333333334\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right), 2 \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right)\right), 1.6666666666666667 + 2 \cdot a\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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))))))))))
      0.9999999999999916)
   (/
    x
    (fma
     y
     (fma
      c
      (*
       2.0
       (fma
        c
        (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))
        (+ a 0.8333333333333334)))
      1.0)
     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)))))))))) <= 0.9999999999999916) {
		tmp = x / fma(y, fma(c, (2.0 * fma(c, ((a + 0.8333333333333334) * (a + 0.8333333333333334)), (a + 0.8333333333333334))), 1.0), 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)))))))))) <= 0.9999999999999916)
		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));
	else
		tmp = 1.0;
	end
	return 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], 0.9999999999999916], 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], 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))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 54.3

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

   8. Simplified 54.3%

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

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. +-lowering-+.f64 N/A

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

       9. +-lowering-+.f64 N/A

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

       10. +-lowering-+.f64 59.8

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

   11. Simplified 59.8%

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

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 0.8× speedup

Math

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

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-24)
   (* (- 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-24) {
		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-24) 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-24) {
		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-24:
		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-24)
		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-24)
		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-24], 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))))))))))) < 4.9999999999999998e-24

   1. Initial program 99.3%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 62.8

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

   5. Simplified 62.8%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 62.3

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 18.1

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

   10. Simplified 18.1%

       \[\leadsto \color{blue}{\frac{x}{y + x}} \]
   11. 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. *-lowering-*.f64 N/A

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

       4. /-lowering-/.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. *-lowering-*.f64 N/A

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

       7. +-commutative N/A

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

       8. +-lowering-+.f64 N/A

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

       9. --lowering--.f64 N/A

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

       10. --lowering--.f64 49.4

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

   12. Applied egg-rr 49.4%

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

   if 4.9999999999999998e-24 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.2%

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

      1. *-inverses 98.2

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 70.8%

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

Alternative 12: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 1.3888888888888888, 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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))))))))))
      0.9999999999999916)
   (/ x (fma y (fma c (fma c 1.3888888888888888 1.6666666666666667) 1.0) 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)))))))))) <= 0.9999999999999916) {
		tmp = x / fma(y, fma(c, fma(c, 1.3888888888888888, 1.6666666666666667), 1.0), 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)))))))))) <= 0.9999999999999916)
		tmp = Float64(x / fma(y, fma(c, fma(c, 1.3888888888888888, 1.6666666666666667), 1.0), x));
	else
		tmp = 1.0;
	end
	return 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], 0.9999999999999916], N[(x / N[(y * N[(c * N[(c * 1.3888888888888888 + 1.6666666666666667), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 49.0

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 49.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(\frac{5}{3} + \frac{25}{18} \cdot c\right)}, x\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(\frac{5}{3} + \frac{25}{18} \cdot c\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \frac{5}{3} + \frac{25}{18} \cdot c, 1\right)}, x\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\frac{25}{18} \cdot c + \frac{5}{3}}, 1\right), x\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{c \cdot \frac{25}{18}} + \frac{5}{3}, 1\right), x\right)} \]

       5. accelerator-lowering-fma.f64 44.5

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, 1.3888888888888888, 1.6666666666666667\right)}, 1\right), x\right)} \]

   14. Simplified 44.5%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, 1.3888888888888888, 1.6666666666666667\right), 1\right)}, x\right)} \]

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 1.3888888888888888, 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(1.6666666666666667, c, 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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))))))))))
      0.9999999999999916)
   (/ x (fma y (fma 1.6666666666666667 c 1.0) 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)))))))))) <= 0.9999999999999916) {
		tmp = x / fma(y, fma(1.6666666666666667, c, 1.0), 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)))))))))) <= 0.9999999999999916)
		tmp = Float64(x / fma(y, fma(1.6666666666666667, c, 1.0), x));
	else
		tmp = 1.0;
	end
	return 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], 0.9999999999999916], N[(x / N[(y * N[(1.6666666666666667 * c + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 49.0

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 49.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 31.0

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(1.6666666666666667, c, 1\right)}, x\right)} \]

   14. Simplified 31.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(1.6666666666666667, c, 1\right)}, x\right)} \]

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(1.6666666666666667, c, 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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))))))))))
      0.9999999999999916)
   (/ 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)))))))))) <= 0.9999999999999916) {
		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)))))))))) <= 0.9999999999999916d0) 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)))))))))) <= 0.9999999999999916) {
		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)))))))))) <= 0.9999999999999916:
		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)))))))))) <= 0.9999999999999916)
		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)))))))))) <= 0.9999999999999916)
		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], 0.9999999999999916], 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))))))))))) < 0.99999999999999156

   1. Initial program 99.3%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 63.0

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

   5. Simplified 63.0%

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

   6. Taylor expanded in b around 0

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

      1. +-lowering-+.f64 18.6

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

   8. Simplified 18.6%

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

   if 0.99999999999999156 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.3%

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

      1. *-inverses 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}} \leq 0.9999999999999916:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.7% accurate, 0.9× speedup

Math

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

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-24)
   (/ 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-24) {
		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-24) 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-24) {
		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-24:
		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-24)
		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-24)
		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-24], 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))))))))))) < 4.9999999999999998e-24

   1. Initial program 99.3%

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

   3. Taylor expanded in b around inf

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

      1. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 62.8

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

   5. Simplified 62.8%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-lowering-*.f64 62.3

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 18.1

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

   10. Simplified 18.1%

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

   11. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 16.8

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

   13. Simplified 16.8%

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

   if 4.9999999999999998e-24 < (/.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 96.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.2%

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

      1. *-inverses 98.2

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 52.4%

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

Alternative 16: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right) \cdot \left(y \cdot b\right), y\right)}\\ \end{array} \end{array} \]

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 -200.0)
     1.0
     (if (<= t_1 4e+288)
       (/
        x
        (fma
         y
         (fma
          c
          (fma
           c
           (fma c 0.7716049382716049 1.3888888888888888)
           1.6666666666666667)
          1.0)
         x))
       (/
        x
        (+
         x
         (fma
          (* 2.0 b)
          (* (* (+ a 0.8333333333333334) (+ a 0.8333333333333334)) (* y b))
          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 <= -200.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+288) {
		tmp = x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x);
	} else {
		tmp = x / (x + fma((2.0 * b), (((a + 0.8333333333333334) * (a + 0.8333333333333334)) * (y * b)), 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 <= -200.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+288)
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x));
	else
		tmp = Float64(x / Float64(x + fma(Float64(2.0 * b), Float64(Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334)) * Float64(y * b)), y)));
	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, -200.0], 1.0, If[LessEqual[t$95$1, 4e+288], N[(x / N[(y * N[(c * N[(c * N[(c * 0.7716049382716049 + 1.3888888888888888), $MachinePrecision] + 1.6666666666666667), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(N[(2.0 * b), $MachinePrecision] * N[(N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] * N[(y * b), $MachinePrecision]), $MachinePrecision] + y), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 4e288

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 74.0

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

   5. Simplified 74.0%

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

   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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 64.4

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

   8. Simplified 64.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 58.2

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 58.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right)}, x\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right), 1\right)}, x\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right) + \frac{5}{3}}, 1\right), x\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, \frac{25}{18} + \frac{125}{162} \cdot c, \frac{5}{3}\right)}, 1\right), x\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\frac{125}{162} \cdot c + \frac{25}{18}}, \frac{5}{3}\right), 1\right), x\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{c \cdot \frac{125}{162}} + \frac{25}{18}, \frac{5}{3}\right), 1\right), x\right)} \]

       7. accelerator-lowering-fma.f64 59.6

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right)}, 1.6666666666666667\right), 1\right), x\right)} \]

   14. Simplified 59.6%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right)}, x\right)} \]

   if 4e288 < (-.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 94.8%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 79.5%

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

   9. Taylor expanded in t around inf

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

       1. /-lowering-/.f64 N/A

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

       2. +-lowering-+.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 64.4%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. +-lowering-+.f64 N/A

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

       8. +-lowering-+.f64 61.9

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

   14. Simplified 61.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right) \cdot \left(y \cdot b\right), y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(b, \left(b \cdot 0.8888888888888888\right) \cdot \frac{y}{t \cdot t}, y\right)}\\ \end{array} \end{array} \]

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 -200.0)
     1.0
     (if (<= t_1 4e+288)
       (/
        x
        (fma
         y
         (fma
          c
          (fma
           c
           (fma c 0.7716049382716049 1.3888888888888888)
           1.6666666666666667)
          1.0)
         x))
       (/ x (+ x (fma b (* (* b 0.8888888888888888) (/ y (* t t))) 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 <= -200.0) {
		tmp = 1.0;
	} else if (t_1 <= 4e+288) {
		tmp = x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x);
	} else {
		tmp = x / (x + fma(b, ((b * 0.8888888888888888) * (y / (t * t))), 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 <= -200.0)
		tmp = 1.0;
	elseif (t_1 <= 4e+288)
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x));
	else
		tmp = Float64(x / Float64(x + fma(b, Float64(Float64(b * 0.8888888888888888) * Float64(y / Float64(t * t))), y)));
	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, -200.0], 1.0, If[LessEqual[t$95$1, 4e+288], N[(x / N[(y * N[(c * N[(c * N[(c * 0.7716049382716049 + 1.3888888888888888), $MachinePrecision] + 1.6666666666666667), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(b * N[(N[(b * 0.8888888888888888), $MachinePrecision] * N[(y / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 4e288

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 74.0

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

   5. Simplified 74.0%

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

   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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 64.4

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

   8. Simplified 64.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 58.2

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 58.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right)}, x\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right), 1\right)}, x\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right) + \frac{5}{3}}, 1\right), x\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, \frac{25}{18} + \frac{125}{162} \cdot c, \frac{5}{3}\right)}, 1\right), x\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\frac{125}{162} \cdot c + \frac{25}{18}}, \frac{5}{3}\right), 1\right), x\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{c \cdot \frac{125}{162}} + \frac{25}{18}, \frac{5}{3}\right), 1\right), x\right)} \]

       7. accelerator-lowering-fma.f64 59.6

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right)}, 1.6666666666666667\right), 1\right), x\right)} \]

   14. Simplified 59.6%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right)}, x\right)} \]

   if 4e288 < (-.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 94.8%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 79.5%

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

   9. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 54.1

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

   11. Simplified 54.1%

       \[\leadsto \frac{x}{x + \mathsf{fma}\left(b, \color{blue}{\frac{0.8888888888888888 \cdot \left(b \cdot y\right)}{t \cdot t}}, y\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. /-lowering-/.f64 N/A

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

       7. *-lowering-*.f64 61.1

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

   13. Applied egg-rr 61.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(b, \left(b \cdot 0.8888888888888888\right) \cdot \frac{y}{t \cdot t}, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\left(b \cdot b\right) \cdot \left(y \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right)\right)}\\ \end{array} \end{array} \]

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 -200.0)
     1.0
     (if (<= t_1 1e+301)
       (/
        x
        (fma
         y
         (fma
          c
          (fma
           c
           (fma c 0.7716049382716049 1.3888888888888888)
           1.6666666666666667)
          1.0)
         x))
       (/
        (* x 0.5)
        (*
         (* b b)
         (* y (* (+ a 0.8333333333333334) (+ 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 <= -200.0) {
		tmp = 1.0;
	} else if (t_1 <= 1e+301) {
		tmp = x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x);
	} else {
		tmp = (x * 0.5) / ((b * b) * (y * ((a + 0.8333333333333334) * (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 <= -200.0)
		tmp = 1.0;
	elseif (t_1 <= 1e+301)
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x));
	else
		tmp = Float64(Float64(x * 0.5) / Float64(Float64(b * b) * Float64(y * Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334)))));
	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, -200.0], 1.0, If[LessEqual[t$95$1, 1e+301], N[(x / N[(y * N[(c * N[(c * N[(c * 0.7716049382716049 + 1.3888888888888888), $MachinePrecision] + 1.6666666666666667), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(y * N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 1.00000000000000005e301

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 63.4

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

   8. Simplified 63.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 57.4

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 57.4%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right)}, x\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right), 1\right)}, x\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right) + \frac{5}{3}}, 1\right), x\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, \frac{25}{18} + \frac{125}{162} \cdot c, \frac{5}{3}\right)}, 1\right), x\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\frac{125}{162} \cdot c + \frac{25}{18}}, \frac{5}{3}\right), 1\right), x\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{c \cdot \frac{125}{162}} + \frac{25}{18}, \frac{5}{3}\right), 1\right), x\right)} \]

       7. accelerator-lowering-fma.f64 58.7

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right)}, 1.6666666666666667\right), 1\right), x\right)} \]

   14. Simplified 58.7%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right)}, x\right)} \]

   if 1.00000000000000005e301 < (-.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 94.6%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in b around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 78.7%

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

   9. Taylor expanded in t around inf

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

       1. /-lowering-/.f64 N/A

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

       2. +-lowering-+.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 65.5%

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

   12. Taylor expanded in b around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. +-lowering-+.f64 N/A

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

       11. +-lowering-+.f64 61.2

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

   14. Simplified 61.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+301}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\left(b \cdot b\right) \cdot \left(y \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, y \cdot \left(b \cdot \left(a \cdot a\right)\right), y\right)}\\ \end{array} \end{array} \]

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 -200.0)
     1.0
     (if (<= t_1 2e+307)
       (/
        x
        (fma
         y
         (fma
          c
          (fma
           c
           (fma c 0.7716049382716049 1.3888888888888888)
           1.6666666666666667)
          1.0)
         x))
       (/ x (+ x (fma (* 2.0 b) (* y (* b (* a a))) 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 <= -200.0) {
		tmp = 1.0;
	} else if (t_1 <= 2e+307) {
		tmp = x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x);
	} else {
		tmp = x / (x + fma((2.0 * b), (y * (b * (a * a))), 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 <= -200.0)
		tmp = 1.0;
	elseif (t_1 <= 2e+307)
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 1.0), x));
	else
		tmp = Float64(x / Float64(x + fma(Float64(2.0 * b), Float64(y * Float64(b * Float64(a * a))), y)));
	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, -200.0], 1.0, If[LessEqual[t$95$1, 2e+307], N[(x / N[(y * N[(c * N[(c * N[(c * 0.7716049382716049 + 1.3888888888888888), $MachinePrecision] + 1.6666666666666667), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(N[(2.0 * b), $MachinePrecision] * N[(y * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 1.99999999999999997e307

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 71.9

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

   5. Simplified 71.9%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 62.1

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

   8. Simplified 62.1%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 56.5

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 56.5%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right)}, x\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right), 1\right)}, x\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right) + \frac{5}{3}}, 1\right), x\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, \frac{25}{18} + \frac{125}{162} \cdot c, \frac{5}{3}\right)}, 1\right), x\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\frac{125}{162} \cdot c + \frac{25}{18}}, \frac{5}{3}\right), 1\right), x\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{c \cdot \frac{125}{162}} + \frac{25}{18}, \frac{5}{3}\right), 1\right), x\right)} \]

       7. accelerator-lowering-fma.f64 57.8

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right)}, 1.6666666666666667\right), 1\right), x\right)} \]

   14. Simplified 57.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right)}, x\right)} \]

   if 1.99999999999999997e307 < (-.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 94.2%

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

   3. Taylor expanded in 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. *-lowering-*.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. --lowering--.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. /-lowering-/.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. +-lowering-+.f64 68.8

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

   5. Simplified 68.8%

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

   6. Taylor expanded in b around 0

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

      1. +-lowering-+.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 78.5%

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

   9. Taylor expanded in t around inf

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

       1. /-lowering-/.f64 N/A

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

       2. +-lowering-+.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

       5. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 64.4%

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

   12. Taylor expanded in a around inf

       \[\leadsto \frac{x}{x + \mathsf{fma}\left(2 \cdot b, \color{blue}{{a}^{2} \cdot \left(b \cdot y\right)}, y\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 57.7

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

   14. Simplified 57.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, y \cdot \left(b \cdot \left(a \cdot a\right)\right), y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(1.6666666666666667, c, 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(a \cdot \left(y \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 -200.0)
     1.0
     (if (<= t_1 5e+218)
       (/ x (fma y (fma 1.6666666666666667 c 1.0) x))
       (/ x (+ x (* 2.0 (* a (* y 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 <= -200.0) {
		tmp = 1.0;
	} else if (t_1 <= 5e+218) {
		tmp = x / fma(y, fma(1.6666666666666667, c, 1.0), x);
	} else {
		tmp = x / (x + (2.0 * (a * (y * 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 <= -200.0)
		tmp = 1.0;
	elseif (t_1 <= 5e+218)
		tmp = Float64(x / fma(y, fma(1.6666666666666667, c, 1.0), x));
	else
		tmp = Float64(x / Float64(x + Float64(2.0 * Float64(a * Float64(y * c)))));
	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, -200.0], 1.0, If[LessEqual[t$95$1, 5e+218], N[(x / N[(y * N[(1.6666666666666667 * c + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(2.0 * N[(a * N[(y * c), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 4.99999999999999983e218

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in 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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 63.9

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

   8. Simplified 63.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 62.2

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 62.2%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 47.3

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(1.6666666666666667, c, 1\right)}, x\right)} \]

   14. Simplified 47.3%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(1.6666666666666667, c, 1\right)}, x\right)} \]

   if 4.99999999999999983e218 < (-.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 95.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 62.8

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

   5. Simplified 62.8%

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

   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. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. +-lowering-+.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 43.8

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

   8. Simplified 43.8%

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

   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. *-lowering-*.f64 N/A

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

       2. *-lowering-*.f64 N/A

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

       3. *-lowering-*.f64 38.2

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

   11. Simplified 38.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(1.6666666666666667, c, 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 2 \cdot \left(a \cdot \left(y \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+241}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(1.6666666666666667, c, 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{a \cdot \left(y \cdot c\right)}\\ \end{array} \end{array} \]

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 -200.0)
     1.0
     (if (<= t_1 1e+241)
       (/ x (fma y (fma 1.6666666666666667 c 1.0) x))
       (/ (* x 0.5) (* a (* y 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 <= -200.0) {
		tmp = 1.0;
	} else if (t_1 <= 1e+241) {
		tmp = x / fma(y, fma(1.6666666666666667, c, 1.0), x);
	} else {
		tmp = (x * 0.5) / (a * (y * 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 <= -200.0)
		tmp = 1.0;
	elseif (t_1 <= 1e+241)
		tmp = Float64(x / fma(y, fma(1.6666666666666667, c, 1.0), x));
	else
		tmp = Float64(Float64(x * 0.5) / Float64(a * Float64(y * c)));
	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, -200.0], 1.0, If[LessEqual[t$95$1, 1e+241], N[(x / N[(y * N[(1.6666666666666667 * c + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] / N[(a * N[(y * c), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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)))))) < 1.0000000000000001e241

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 70.0

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

   5. Simplified 70.0%

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

   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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 64.0

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

   8. Simplified 64.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 62.5

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 62.5%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 45.1

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(1.6666666666666667, c, 1\right)}, x\right)} \]

   14. Simplified 45.1%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(1.6666666666666667, c, 1\right)}, x\right)} \]

   if 1.0000000000000001e241 < (-.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 95.7%

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

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate--l+ N/A

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

      8. +-lowering-+.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 45.1

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

   8. Simplified 45.1%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.7

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

   11. Simplified 36.7%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+241}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(1.6666666666666667, c, 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{a \cdot \left(y \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \end{array} \end{array} \]

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)))))
      -200.0)
   1.0
   (/
    x
    (fma
     y
     (fma
      c
      (fma c (fma c 0.7716049382716049 1.3888888888888888) 1.6666666666666667)
      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))))) <= -200.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 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))))) <= -200.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(c, 0.7716049382716049, 1.3888888888888888), 1.6666666666666667), 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], -200.0], 1.0, N[(x / N[(y * N[(c * N[(c * N[(c * 0.7716049382716049 + 1.3888888888888888), $MachinePrecision] + 1.6666666666666667), $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)))))) < -200

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

      1. *-inverses 99.1

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

   7. Applied egg-rr 99.1%

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

   if -200 < (-.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 97.4%

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

   3. Taylor expanded in 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. *-lowering-*.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. +-lowering-+.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. +-lowering-+.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. /-lowering-/.f64 65.5

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

   5. Simplified 65.5%

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

   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. accelerator-lowering-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. exp-lowering-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. *-lowering-*.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. *-lowering-*.f64 N/A

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

      6. +-lowering-+.f64 56.9

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

   8. Simplified 56.9%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{\frac{5}{3} \cdot c}}, x\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 51.3

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   11. Simplified 51.3%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{\color{blue}{1.6666666666666667 \cdot c}}, x\right)} \]

   12. Taylor expanded in c around 0

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right)}, x\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{c \cdot \left(\frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right)\right) + 1}, x\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \frac{5}{3} + c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right), 1\right)}, x\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{c \cdot \left(\frac{25}{18} + \frac{125}{162} \cdot c\right) + \frac{5}{3}}, 1\right), x\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, \frac{25}{18} + \frac{125}{162} \cdot c, \frac{5}{3}\right)}, 1\right), x\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\frac{125}{162} \cdot c + \frac{25}{18}}, \frac{5}{3}\right), 1\right), x\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{c \cdot \frac{125}{162}} + \frac{25}{18}, \frac{5}{3}\right), 1\right), x\right)} \]

       7. accelerator-lowering-fma.f64 53.3

          \[\leadsto \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right)}, 1.6666666666666667\right), 1\right), x\right)} \]

   14. Simplified 53.3%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right)}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(c, 0.7716049382716049, 1.3888888888888888\right), 1.6666666666666667\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 51.2% accurate, 198.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around inf

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

5. Simplified 45.1%

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

   1. *-inverses 45.1

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

7. Applied egg-rr 45.1%

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

Developer Target 1: 94.9% 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 2024195 
(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)))))))))))