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

Percentage Accurate: 93.9% → 93.9%

Time: 20.2s

Alternatives: 24

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: 93.9% 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: 93.9% 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 97.6%

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

3. Final simplification 97.6%

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

Alternative 2: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(a, 2 \cdot \left(c + a \cdot \left(c \cdot c\right)\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          y
          (exp
           (*
            2.0
            (+
             (/ (* z (sqrt (+ t a))) t)
             (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
   (if (<= t_1 -1e+248)
     (* (- y x) (/ x (* (+ x y) (- y x))))
     (if (<= t_1 0.0)
       1.0
       (/ x (+ x (* y (fma a (* 2.0 (+ c (* a (* c c)))) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))));
	double tmp;
	if (t_1 <= -1e+248) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else if (t_1 <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * fma(a, (2.0 * (c + (a * (c * c)))), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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))))))))
	tmp = 0.0
	if (t_1 <= -1e+248)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	elseif (t_1 <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(a, Float64(2.0 * Float64(c + Float64(a * Float64(c * c)))), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+248], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], 1.0, N[(x / N[(x + N[(y * N[(a * N[(2.0 * N[(c + N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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))))))))) < -1.00000000000000005e248

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 60.2

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

   5. Simplified 60.2%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 15.6

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

   8. Simplified 15.6%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 57.2

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

   10. Applied egg-rr 57.2%

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

   if -1.00000000000000005e248 < (*.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))))))))) < -0.0

   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 -0.0 < (*.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 94.7%

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

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 44.5

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

   8. Simplified 44.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

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

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

       3. distribute-lft-out N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 55.6

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

   11. Simplified 55.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{elif}\;y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(a, 2 \cdot \left(c + a \cdot \left(c \cdot c\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b \cdot -2, a + 0.8333333333333334, 1\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          y
          (exp
           (*
            2.0
            (+
             (/ (* z (sqrt (+ t a))) t)
             (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))))))
   (if (<= t_1 -1e+248)
     (* (- y x) (/ x (* (+ x y) (- y x))))
     (if (<= t_1 0.0)
       1.0
       (/ x (fma y (fma (* b -2.0) (+ a 0.8333333333333334) 1.0) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))));
	double tmp;
	if (t_1 <= -1e+248) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else if (t_1 <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(y, fma((b * -2.0), (a + 0.8333333333333334), 1.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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))))))))
	tmp = 0.0
	if (t_1 <= -1e+248)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	elseif (t_1 <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(y, fma(Float64(b * -2.0), Float64(a + 0.8333333333333334), 1.0), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+248], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], 1.0, N[(x / N[(y * N[(N[(b * -2.0), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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))))))))) < -1.00000000000000005e248

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 60.2

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

   5. Simplified 60.2%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 15.6

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

   8. Simplified 15.6%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 57.2

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

   10. Applied egg-rr 57.2%

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

   if -1.00000000000000005e248 < (*.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))))))))) < -0.0

   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 -0.0 < (*.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 94.7%

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

   3. Taylor expanded in b around inf

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

      1. *-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 68.2

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

   5. Simplified 68.2%

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

   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 57.3

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

   8. Simplified 57.3%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. +-lowering-+.f64 47.7

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

   11. Simplified 47.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{elif}\;y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b \cdot -2, a + 0.8333333333333334, 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))))
   (if (<= t_1 0.0)
     1.0
     (if (<= t_1 2.0) (/ x (+ x y)) (* (- y x) (/ x (* (+ x y) (- y x))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0;
	} else if (t_1 <= 2.0) {
		tmp = x / (x + y);
	} else {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((2.0d0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0)))))))
    if (t_1 <= 0.0d0) then
        tmp = 1.0d0
    else if (t_1 <= 2.0d0) then
        tmp = x / (x + y)
    else
        tmp = (y - x) * (x / ((x + y) * (y - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0;
	} else if (t_1 <= 2.0) {
		tmp = x / (x + y);
	} else {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = math.exp((2.0 * (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0)))))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = 1.0
	elif t_1 <= 2.0:
		tmp = x / (x + y)
	else:
		tmp = (y - x) * (x / ((x + y) * (y - x)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 0.0], 1.0, If[LessEqual[t$95$1, 2.0], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (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)))))))) < 0.0

   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 0.0 < (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)))))))) < 2

   1. Initial program 99.2%

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

   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 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

   if 2 < (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 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 63.3

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

   5. Simplified 63.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 13.9

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

   8. Simplified 13.9%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 43.4

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

   10. Applied egg-rr 43.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{elif}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq 2:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ t_2 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+281}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \sqrt{\frac{1}{t}}, 0.6666666666666666 \cdot \frac{b - c}{t}\right) + \left(a + 0.8333333333333334\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, t\_2 \cdot t\_2, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

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)))))
   (if (<= t_1 -5e+80)
     1.0
     (if (<= t_1 1e+281)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (+
             (fma z (sqrt (/ 1.0 t)) (* 0.6666666666666666 (/ (- b c) t)))
             (* (+ a 0.8333333333333334) (- c b))))))))
       (/ x (+ x (* y (fma c (* 2.0 (fma c (* t_2 t_2) t_2)) 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 tmp;
	if (t_1 <= -5e+80) {
		tmp = 1.0;
	} else if (t_1 <= 1e+281) {
		tmp = x / (x + (y * exp((2.0 * (fma(z, sqrt((1.0 / t)), (0.6666666666666666 * ((b - c) / t))) + ((a + 0.8333333333333334) * (c - b)))))));
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * fma(c, (t_2 * t_2), t_2)), 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)))
	tmp = 0.0
	if (t_1 <= -5e+80)
		tmp = 1.0;
	elseif (t_1 <= 1e+281)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(z, sqrt(Float64(1.0 / t)), Float64(0.6666666666666666 * Float64(Float64(b - c) / t))) + Float64(Float64(a + 0.8333333333333334) * Float64(c - b))))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(2.0 * fma(c, Float64(t_2 * t_2), t_2)), 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]}, If[LessEqual[t$95$1, -5e+80], 1.0, If[LessEqual[t$95$1, 1e+281], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(z * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] + N[(0.6666666666666666 * N[(N[(b - c), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(c * N[(2.0 * N[(c * N[(t$95$2 * t$95$2), $MachinePrecision] + t$95$2), $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)))))) < -4.99999999999999961e80

   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 -4.99999999999999961e80 < (-.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)))))) < 1e281

   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 t around inf

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 88.8

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

   5. Simplified 88.8%

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

   if 1e281 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 93.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 89.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -5 \cdot 10^{+80}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+281}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \sqrt{\frac{1}{t}}, 0.6666666666666666 \cdot \frac{b - c}{t}\right) + \left(a + 0.8333333333333334\right) \cdot \left(c - b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ t_2 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 10^{+159}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t))))
        (t_2
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_2 -2e+17)
     1.0
     (if (<= t_2 1e+159)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* c (+ a (+ 0.8333333333333334 (/ -0.6666666666666666 t)))))))))
       (/ x (+ x (* y (fma c (* 2.0 (fma c (* 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 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	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 <= -2e+17) {
		tmp = 1.0;
	} else if (t_2 <= 1e+159) {
		tmp = x / (x + (y * exp((2.0 * (c * (a + (0.8333333333333334 + (-0.6666666666666666 / t))))))));
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * fma(c, (t_1 * t_1), t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	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 <= -2e+17)
		tmp = 1.0;
	elseif (t_2 <= 1e+159)
		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(c, Float64(2.0 * fma(c, 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[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $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, -2e+17], 1.0, If[LessEqual[t$95$2, 1e+159], 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[(c * N[(2.0 * N[(c * 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)))))) < -2e17

   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 -2e17 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 9.9999999999999993e158

   1. Initial program 99.6%

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

   3. Taylor expanded in c around inf

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

      1. *-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 81.6

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

   5. Simplified 81.6%

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

   if 9.9999999999999993e158 < (-.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 95.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 64.6

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

   5. Simplified 64.6%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 84.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+159}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + \left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ t_2 := \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+207}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(2, t\_2, \left(\left(a + 0.8333333333333334\right) \cdot t\_2\right) \cdot \left(c \cdot 1.3333333333333333\right)\right), \mathsf{fma}\left(2, a, 1.6666666666666667\right)\right), 1\right), x\right)}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, a \cdot \mathsf{fma}\left(y, \frac{\frac{0.6666666666666666}{t} + -0.8333333333333334}{a}, -y\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(2 \cdot b, t\_2, -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))
        (t_2 (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 1e+207)
       (/
        x
        (fma
         y
         (fma
          c
          (fma
           c
           (fma
            2.0
            t_2
            (* (* (+ a 0.8333333333333334) t_2) (* c 1.3333333333333333)))
           (fma 2.0 a 1.6666666666666667))
          1.0)
         x))
       (if (<= t_1 4e+279)
         (/
          x
          (+
           x
           (fma
            (* 2.0 b)
            (*
             a
             (fma
              y
              (/ (+ (/ 0.6666666666666666 t) -0.8333333333333334) a)
              (- y)))
            y)))
         (/
          x
          (fma
           y
           (fma b (fma (* 2.0 b) t_2 (+ -1.6666666666666667 (* a -2.0))) 1.0)
           x)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double t_2 = (a + 0.8333333333333334) * (a + 0.8333333333333334);
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 1e+207) {
		tmp = x / fma(y, fma(c, fma(c, fma(2.0, t_2, (((a + 0.8333333333333334) * t_2) * (c * 1.3333333333333333))), fma(2.0, a, 1.6666666666666667)), 1.0), x);
	} else if (t_1 <= 4e+279) {
		tmp = x / (x + fma((2.0 * b), (a * fma(y, (((0.6666666666666666 / t) + -0.8333333333333334) / a), -y)), y));
	} else {
		tmp = x / fma(y, fma(b, fma((2.0 * b), t_2, (-1.6666666666666667 + (a * -2.0))), 1.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	t_2 = Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 1e+207)
		tmp = Float64(x / fma(y, fma(c, fma(c, fma(2.0, t_2, Float64(Float64(Float64(a + 0.8333333333333334) * t_2) * Float64(c * 1.3333333333333333))), fma(2.0, a, 1.6666666666666667)), 1.0), x));
	elseif (t_1 <= 4e+279)
		tmp = Float64(x / Float64(x + fma(Float64(2.0 * b), Float64(a * fma(y, Float64(Float64(Float64(0.6666666666666666 / t) + -0.8333333333333334) / a), Float64(-y))), y)));
	else
		tmp = Float64(x / fma(y, fma(b, fma(Float64(2.0 * b), t_2, Float64(-1.6666666666666667 + Float64(a * -2.0))), 1.0), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 1e+207], N[(x / N[(y * N[(c * N[(c * N[(2.0 * t$95$2 + N[(N[(N[(a + 0.8333333333333334), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(c * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * a + 1.6666666666666667), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+279], N[(x / N[(x + N[(N[(2.0 * b), $MachinePrecision] * N[(a * N[(y * N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] + -0.8333333333333334), $MachinePrecision] / a), $MachinePrecision] + (-y)), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(b * N[(N[(2.0 * b), $MachinePrecision] * t$95$2 + N[(-1.6666666666666667 + N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 1e207

   1. Initial program 99.7%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 76.3

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

   5. Simplified 76.3%

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

   6. Taylor expanded in 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 68.7

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

   8. Simplified 68.7%

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

   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 65.7%

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

   if 1e207 < (-.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)))))) < 4.00000000000000023e279

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 72.9

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

   5. Simplified 72.9%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-lowering-+.f64 58.1

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

   8. Simplified 58.1%

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

   9. Taylor expanded in a around inf

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

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

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

       2. +-commutative N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

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

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. /-lowering-/.f64 N/A

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 73.3

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

   11. Simplified 73.3%

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

   if 4.00000000000000023e279 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 93.9%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 64.2

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

   5. Simplified 64.2%

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

   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 52.4

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

   8. Simplified 52.4%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

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

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

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

       14. *-lowering-*.f64 72.6

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

   11. Simplified 72.6%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(2 \cdot b, \left(0.8333333333333334 + a\right) \cdot \left(0.8333333333333334 + a\right), -1.6666666666666667 + -2 \cdot a\right), 1\right)}, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+207}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(2, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), \left(\left(a + 0.8333333333333334\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right)\right) \cdot \left(c \cdot 1.3333333333333333\right)\right), \mathsf{fma}\left(2, a, 1.6666666666666667\right)\right), 1\right), x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 4 \cdot 10^{+279}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, a \cdot \mathsf{fma}\left(y, \frac{\frac{0.6666666666666666}{t} + -0.8333333333333334}{a}, -y\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(2 \cdot b, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ t_2 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 10^{+159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{-2 \cdot \left(b \cdot \left(a + 0.8333333333333334\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, t\_1 \cdot t\_1, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t))))
        (t_2
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_2 -1e+63)
     1.0
     (if (<= t_2 1e+159)
       (/ x (fma y (exp (* -2.0 (* b (+ a 0.8333333333333334)))) x))
       (/ x (+ x (* y (fma c (* 2.0 (fma c (* 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 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	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 <= -1e+63) {
		tmp = 1.0;
	} else if (t_2 <= 1e+159) {
		tmp = x / fma(y, exp((-2.0 * (b * (a + 0.8333333333333334)))), x);
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * fma(c, (t_1 * t_1), t_1)), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	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 <= -1e+63)
		tmp = 1.0;
	elseif (t_2 <= 1e+159)
		tmp = Float64(x / fma(y, exp(Float64(-2.0 * Float64(b * Float64(a + 0.8333333333333334)))), x));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(2.0 * fma(c, 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[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $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, -1e+63], 1.0, If[LessEqual[t$95$2, 1e+159], N[(x / N[(y * N[Exp[N[(-2.0 * N[(b * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(c * N[(2.0 * N[(c * 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)))))) < -1.00000000000000006e63

   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 -1.00000000000000006e63 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 9.9999999999999993e158

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-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 76.4

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

   5. Simplified 76.4%

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

   6. Taylor expanded in 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 76.4

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

   8. Simplified 76.4%

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

   if 9.9999999999999993e158 < (-.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 95.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 64.6

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

   5. Simplified 64.6%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 84.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -1 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{-2 \cdot \left(b \cdot \left(a + 0.8333333333333334\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ t_2 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (/
          x
          (fma
           y
           (fma
            c
            (*
             2.0
             (fma
              c
              (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))
              (+ a 0.8333333333333334)))
            1.0)
           x)))
        (t_2
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_2 -2e+17)
     1.0
     (if (<= t_2 0.1)
       t_1
       (if (<= t_2 2e+219) (* (- y x) (/ x (* (+ x y) (- y x)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x / fma(y, fma(c, (2.0 * fma(c, ((a + 0.8333333333333334) * (a + 0.8333333333333334)), (a + 0.8333333333333334))), 1.0), x);
	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 <= -2e+17) {
		tmp = 1.0;
	} else if (t_2 <= 0.1) {
		tmp = t_1;
	} else if (t_2 <= 2e+219) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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))
	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 <= -2e+17)
		tmp = 1.0;
	elseif (t_2 <= 0.1)
		tmp = t_1;
	elseif (t_2 <= 2e+219)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, 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, -2e+17], 1.0, If[LessEqual[t$95$2, 0.1], t$95$1, If[LessEqual[t$95$2, 2e+219], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 0.10000000000000001 or 1.99999999999999993e219 < (-.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 95.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 71.4

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

   5. Simplified 71.4%

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

   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 55.6

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

   8. Simplified 55.6%

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

   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 72.0

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

   11. Simplified 72.0%

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

   if 0.10000000000000001 < (-.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)))))) < 1.99999999999999993e219

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 12.2

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

   8. Simplified 12.2%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 53.2

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

   10. Applied egg-rr 53.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 0.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{elif}\;t\_1 \leq 1.2 \cdot 10^{+219}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \left(a + c \cdot \left(a \cdot a\right)\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 0.1)
       (/ x (fma y (fma 2.0 (* c (+ a 0.8333333333333334)) 1.0) x))
       (if (<= t_1 1.2e+219)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (/ x (+ x (* y (fma c (* 2.0 (+ a (* c (* a a)))) 1.0)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 0.1) {
		tmp = x / fma(y, fma(2.0, (c * (a + 0.8333333333333334)), 1.0), x);
	} else if (t_1 <= 1.2e+219) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * (a + (c * (a * a)))), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 0.1)
		tmp = Float64(x / fma(y, fma(2.0, Float64(c * Float64(a + 0.8333333333333334)), 1.0), x));
	elseif (t_1 <= 1.2e+219)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(2.0 * Float64(a + Float64(c * Float64(a * a)))), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 0.1], N[(x / N[(y * N[(2.0 * N[(c * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.2e+219], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(c * N[(2.0 * N[(a + N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e17

   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 -2e17 < (-.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)))))) < 0.10000000000000001

   1. Initial program 99.2%

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

   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 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. 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 94.8

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

   8. Simplified 94.8%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. +-lowering-+.f64 90.9

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

   11. Simplified 90.9%

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

   if 0.10000000000000001 < (-.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)))))) < 1.2e219

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 57.3

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

   5. Simplified 57.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 12.4

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

   8. Simplified 12.4%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 54.6

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

   10. Applied egg-rr 54.6%

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

   if 1.2e219 < (-.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 95.2%

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

   3. Taylor expanded in c around inf

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

      1. *-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 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 41.1

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

   8. Simplified 41.1%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

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

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

       3. distribute-lft-out N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 59.7

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

   11. Simplified 59.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 0.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 1.2 \cdot 10^{+219}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \left(a + c \cdot \left(a \cdot a\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{elif}\;t\_1 \leq 1.2 \cdot 10^{+219}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x + y, t, 1.3333333333333333 \cdot \left(y \cdot b\right)\right)}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 0.1)
       (/ x (fma y (fma 2.0 (* c (+ a 0.8333333333333334)) 1.0) x))
       (if (<= t_1 1.2e+219)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (/ x (/ (fma (+ x y) t (* 1.3333333333333333 (* y b))) t)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 0.1) {
		tmp = x / fma(y, fma(2.0, (c * (a + 0.8333333333333334)), 1.0), x);
	} else if (t_1 <= 1.2e+219) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (fma((x + y), t, (1.3333333333333333 * (y * b))) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 0.1)
		tmp = Float64(x / fma(y, fma(2.0, Float64(c * Float64(a + 0.8333333333333334)), 1.0), x));
	elseif (t_1 <= 1.2e+219)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = Float64(x / Float64(fma(Float64(x + y), t, Float64(1.3333333333333333 * Float64(y * b))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 0.1], N[(x / N[(y * N[(2.0 * N[(c * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.2e+219], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(x + y), $MachinePrecision] * t + N[(1.3333333333333333 * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 0.10000000000000001

   1. Initial program 99.2%

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

   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 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. 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 94.8

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

   8. Simplified 94.8%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. +-lowering-+.f64 90.9

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

   11. Simplified 90.9%

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

   if 0.10000000000000001 < (-.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)))))) < 1.2e219

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 57.3

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

   5. Simplified 57.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 12.4

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

   8. Simplified 12.4%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 54.6

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

   10. Applied egg-rr 54.6%

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

   if 1.2e219 < (-.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 95.2%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 65.2

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

   5. Simplified 65.2%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-lowering-+.f64 54.5

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

   8. Simplified 54.5%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + \mathsf{fma}\left(2 \cdot b, \color{blue}{\frac{2}{3} \cdot \frac{y}{t}}, y\right)} \]
   10. Step-by-step derivation

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

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

       2. /-lowering-/.f64 43.3

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

   11. Simplified 43.3%

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

   12. Taylor expanded in t around 0

       \[\leadsto \frac{x}{\color{blue}{\frac{\frac{4}{3} \cdot \left(b \cdot y\right) + t \cdot \left(x + y\right)}{t}}} \]
   13. Step-by-step derivation

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

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \frac{x}{\frac{\color{blue}{\mathsf{fma}\left(x + y, t, \frac{4}{3} \cdot \left(b \cdot y\right)\right)}}{t}} \]

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

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(\color{blue}{x + y}, t, \frac{4}{3} \cdot \left(b \cdot y\right)\right)}{t}} \]

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

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, t, \color{blue}{\frac{4}{3} \cdot \left(b \cdot y\right)}\right)}{t}} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, t, \frac{4}{3} \cdot \color{blue}{\left(y \cdot b\right)}\right)}{t}} \]

       8. *-lowering-*.f64 49.8

          \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(x + y, t, 1.3333333333333333 \cdot \color{blue}{\left(y \cdot b\right)}\right)}{t}} \]

   14. Simplified 49.8%

       \[\leadsto \frac{x}{\color{blue}{\frac{\mathsf{fma}\left(x + y, t, 1.3333333333333333 \cdot \left(y \cdot b\right)\right)}{t}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 0.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 1.2 \cdot 10^{+219}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(x + y, t, 1.3333333333333333 \cdot \left(y \cdot b\right)\right)}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ t_2 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ x (fma y (fma 2.0 (* c (+ a 0.8333333333333334)) 1.0) x)))
        (t_2
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_2 -2e+17)
     1.0
     (if (<= t_2 0.1)
       t_1
       (if (<= t_2 5e+293) (* (- y x) (/ x (* (+ x y) (- y x)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x / fma(y, fma(2.0, (c * (a + 0.8333333333333334)), 1.0), x);
	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 <= -2e+17) {
		tmp = 1.0;
	} else if (t_2 <= 0.1) {
		tmp = t_1;
	} else if (t_2 <= 5e+293) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(x / fma(y, fma(2.0, Float64(c * Float64(a + 0.8333333333333334)), 1.0), x))
	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 <= -2e+17)
		tmp = 1.0;
	elseif (t_2 <= 0.1)
		tmp = t_1;
	elseif (t_2 <= 5e+293)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x / N[(y * N[(2.0 * N[(c * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $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, -2e+17], 1.0, If[LessEqual[t$95$2, 0.1], t$95$1, If[LessEqual[t$95$2, 5e+293], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 0.10000000000000001 or 5.00000000000000033e293 < (-.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 94.5%

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

   3. Taylor expanded in c around inf

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

      1. *-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 75.7

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

   5. Simplified 75.7%

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

   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 60.1

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

   8. Simplified 60.1%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. +-lowering-+.f64 58.6

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

   11. Simplified 58.6%

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

   if 0.10000000000000001 < (-.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)))))) < 5.00000000000000033e293

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 61.6

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

   5. Simplified 61.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 15.1

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

   8. Simplified 15.1%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 48.8

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

   10. Applied egg-rr 48.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 0.1:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, c \cdot \left(a + 0.8333333333333334\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(2, a \cdot c, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 0.1)
       (/ x (+ x y))
       (if (<= t_1 5e+293)
         (* (- y x) (/ x (* (+ x y) (- y x))))
         (/ x (+ x (* y (fma 2.0 (* a c) 1.0)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 0.1) {
		tmp = x / (x + y);
	} else if (t_1 <= 5e+293) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = x / (x + (y * fma(2.0, (a * c), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 0.1)
		tmp = Float64(x / Float64(x + y));
	elseif (t_1 <= 5e+293)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(2.0, Float64(a * c), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 0.1], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+293], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(2.0 * N[(a * c), $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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 0.10000000000000001

   1. Initial program 99.2%

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

   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 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 90.5

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

   8. Simplified 90.5%

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

   if 0.10000000000000001 < (-.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)))))) < 5.00000000000000033e293

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 61.6

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

   5. Simplified 61.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 15.1

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

   8. Simplified 15.1%

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

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. *-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 48.8

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

   10. Applied egg-rr 48.8%

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

   if 5.00000000000000033e293 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 93.3%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 69.9

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

   5. Simplified 69.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 49.1

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

   8. Simplified 49.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

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

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

       3. *-lowering-*.f64 48.1

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

   11. Simplified 48.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 0.1:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(2, a \cdot c, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ t_2 := \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\\ t_3 := 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(2, t\_2, \left(b \cdot -1.3333333333333333\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot t\_2\right)\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, t\_3 \cdot t\_3, t\_3\right), 1\right)}\\ \end{array} \end{array} \]

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 (* (+ a 0.8333333333333334) (+ a 0.8333333333333334)))
        (t_3 (+ 0.8333333333333334 (- a (/ 0.6666666666666666 t)))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 1e+159)
       (/
        x
        (fma
         y
         (fma
          b
          (fma
           b
           (fma
            2.0
            t_2
            (* (* b -1.3333333333333333) (* (+ a 0.8333333333333334) t_2)))
           (+ -1.6666666666666667 (* a -2.0)))
          1.0)
         x))
       (/ x (+ x (* y (fma c (* 2.0 (fma c (* 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 = (a + 0.8333333333333334) * (a + 0.8333333333333334);
	double t_3 = 0.8333333333333334 + (a - (0.6666666666666666 / t));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 1e+159) {
		tmp = x / fma(y, fma(b, fma(b, fma(2.0, t_2, ((b * -1.3333333333333333) * ((a + 0.8333333333333334) * t_2))), (-1.6666666666666667 + (a * -2.0))), 1.0), x);
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * fma(c, (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(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334))
	t_3 = Float64(0.8333333333333334 + Float64(a - Float64(0.6666666666666666 / t)))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 1e+159)
		tmp = Float64(x / fma(y, fma(b, fma(b, fma(2.0, t_2, Float64(Float64(b * -1.3333333333333333) * Float64(Float64(a + 0.8333333333333334) * t_2))), Float64(-1.6666666666666667 + Float64(a * -2.0))), 1.0), x));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(2.0 * fma(c, 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[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.8333333333333334 + N[(a - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 1e+159], N[(x / N[(y * N[(b * N[(b * N[(2.0 * t$95$2 + N[(N[(b * -1.3333333333333333), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.6666666666666667 + N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(c * N[(2.0 * N[(c * N[(t$95$3 * t$95$3), $MachinePrecision] + t$95$3), $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)))))) < -2e17

   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 -2e17 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 9.9999999999999993e158

   1. Initial program 99.6%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 74.5

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

   5. Simplified 74.5%

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

   6. Taylor expanded in 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 74.5

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

   8. Simplified 74.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

   11. Simplified 71.9%

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

   if 9.9999999999999993e158 < (-.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 95.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 64.6

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

   5. Simplified 64.6%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 84.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+159}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(2, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), \left(b \cdot -1.3333333333333333\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right)\right)\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right) \cdot \left(0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 0.8333333333333334 + \left(a - \frac{0.6666666666666666}{t}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ t_2 := \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+281}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(2, t\_2, \left(b \cdot -1.3333333333333333\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot t\_2\right)\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, t\_2, a + 0.8333333333333334\right), 1\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0))))))
        (t_2 (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 1e+281)
       (/
        x
        (fma
         y
         (fma
          b
          (fma
           b
           (fma
            2.0
            t_2
            (* (* b -1.3333333333333333) (* (+ a 0.8333333333333334) t_2)))
           (+ -1.6666666666666667 (* a -2.0)))
          1.0)
         x))
       (/
        x
        (fma y (fma c (* 2.0 (fma c t_2 (+ a 0.8333333333333334))) 1.0) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double t_2 = (a + 0.8333333333333334) * (a + 0.8333333333333334);
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 1e+281) {
		tmp = x / fma(y, fma(b, fma(b, fma(2.0, t_2, ((b * -1.3333333333333333) * ((a + 0.8333333333333334) * t_2))), (-1.6666666666666667 + (a * -2.0))), 1.0), x);
	} else {
		tmp = x / fma(y, fma(c, (2.0 * fma(c, t_2, (a + 0.8333333333333334))), 1.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	t_2 = Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 1e+281)
		tmp = Float64(x / fma(y, fma(b, fma(b, fma(2.0, t_2, Float64(Float64(b * -1.3333333333333333) * Float64(Float64(a + 0.8333333333333334) * t_2))), Float64(-1.6666666666666667 + Float64(a * -2.0))), 1.0), x));
	else
		tmp = Float64(x / fma(y, fma(c, Float64(2.0 * fma(c, t_2, Float64(a + 0.8333333333333334))), 1.0), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 1e+281], N[(x / N[(y * N[(b * N[(b * N[(2.0 * t$95$2 + N[(N[(b * -1.3333333333333333), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.6666666666666667 + N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(c * N[(2.0 * N[(c * t$95$2 + N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -2e17

   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 -2e17 < (-.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)))))) < 1e281

   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 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 69.8

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

   5. Simplified 69.8%

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

   6. Taylor expanded in 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 62.2

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

   8. Simplified 62.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

   11. Simplified 64.8%

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

   if 1e281 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 93.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in 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 52.4

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

   8. Simplified 52.4%

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

   9. Taylor expanded in 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 73.0

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

   11. Simplified 73.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+281}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(2, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), \left(b \cdot -1.3333333333333333\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(\left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right)\right)\right)\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), a + 0.8333333333333334\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(2 \cdot b, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (exp
       (*
        2.0
        (+
         (/ (* z (sqrt (+ t a))) t)
         (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
      0.0)
   1.0
   (/
    x
    (fma
     y
     (fma
      b
      (fma
       (* 2.0 b)
       (* (+ a 0.8333333333333334) (+ a 0.8333333333333334))
       (+ -1.6666666666666667 (* a -2.0)))
      1.0)
     x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))))) <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(y, fma(b, fma((2.0 * b), ((a + 0.8333333333333334) * (a + 0.8333333333333334)), (-1.6666666666666667 + (a * -2.0))), 1.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (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))))))) <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(y, fma(b, fma(Float64(2.0 * b), Float64(Float64(a + 0.8333333333333334) * Float64(a + 0.8333333333333334)), Float64(-1.6666666666666667 + Float64(a * -2.0))), 1.0), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[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], 0.0], 1.0, N[(x / N[(y * N[(b * N[(N[(2.0 * b), $MachinePrecision] * N[(N[(a + 0.8333333333333334), $MachinePrecision] * N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision] + N[(-1.6666666666666667 + N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (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)))))))) < 0.0

   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 0.0 < (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 96.7%

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

   3. Taylor expanded in b around inf

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

      1. *-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 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in 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 56.6

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

   8. Simplified 56.6%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

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

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

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

       14. *-lowering-*.f64 65.1

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

   11. Simplified 65.1%

       \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(2 \cdot b, \left(0.8333333333333334 + a\right) \cdot \left(0.8333333333333334 + a\right), -1.6666666666666667 + -2 \cdot a\right), 1\right)}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)} \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(2 \cdot b, \left(a + 0.8333333333333334\right) \cdot \left(a + 0.8333333333333334\right), -1.6666666666666667 + a \cdot -2\right), 1\right), x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.5% accurate, 0.9× speedup

Math

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

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))))))))))
      4e-21)
   (/ 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)))))))))) <= 4e-21) {
		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)))))))))) <= 4d-21) 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)))))))))) <= 4e-21) {
		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)))))))))) <= 4e-21:
		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)))))))))) <= 4e-21)
		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)))))))))) <= 4e-21)
		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], 4e-21], 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))))))))))) < 3.99999999999999963e-21

   1. Initial program 99.2%

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

   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 64.2

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

   5. Simplified 64.2%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 17.3

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

   8. Simplified 17.3%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 16.1

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

   11. Simplified 16.1%

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

   if 3.99999999999999963e-21 < (/.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 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 x around inf

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

   5. Simplified 95.8%

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

      1. *-inverses 95.8

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

   7. Applied egg-rr 95.8%

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

4. Final simplification 51.9%

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

Alternative 18: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+281}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, y \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \left(a + c \cdot \left(a \cdot a\right)\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 1e+281)
       (/
        x
        (+
         x
         (fma
          (* 2.0 b)
          (* y (- (/ 0.6666666666666666 t) (+ a 0.8333333333333334)))
          y)))
       (/ x (+ x (* y (fma c (* 2.0 (+ a (* c (* a a)))) 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 1e+281) {
		tmp = x / (x + fma((2.0 * b), (y * ((0.6666666666666666 / t) - (a + 0.8333333333333334))), y));
	} else {
		tmp = x / (x + (y * fma(c, (2.0 * (a + (c * (a * a)))), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 1e+281)
		tmp = Float64(x / Float64(x + fma(Float64(2.0 * b), Float64(y * Float64(Float64(0.6666666666666666 / t) - Float64(a + 0.8333333333333334))), y)));
	else
		tmp = Float64(x / Float64(x + Float64(y * fma(c, Float64(2.0 * Float64(a + Float64(c * Float64(a * a)))), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 1e+281], N[(x / N[(x + N[(N[(2.0 * b), $MachinePrecision] * N[(y * N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(a + 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[(c * N[(2.0 * N[(a + N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 1e281

   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 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 69.8

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

   5. Simplified 69.8%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-lowering-+.f64 54.3

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

   8. Simplified 54.3%

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

   if 1e281 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))

   1. Initial program 93.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 47.6

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

   8. Simplified 47.6%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

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

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

       3. distribute-lft-out N/A

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

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

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 65.8

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

   11. Simplified 65.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+281}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(2 \cdot b, y \cdot \left(\frac{0.6666666666666666}{t} - \left(a + 0.8333333333333334\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot \mathsf{fma}\left(c, 2 \cdot \left(a + c \cdot \left(a \cdot a\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+226}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \frac{x}{y}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + -2 \cdot \left(a \cdot \left(y \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 1e+226)
       (/ x (fma y (/ x y) y))
       (/ x (+ x (* -2.0 (* a (* y b)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 1e+226) {
		tmp = x / fma(y, (x / y), y);
	} else {
		tmp = x / (x + (-2.0 * (a * (y * b))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 1e+226)
		tmp = Float64(x / fma(y, Float64(x / y), y));
	else
		tmp = Float64(x / Float64(x + Float64(-2.0 * Float64(a * Float64(y * b)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 1e+226], N[(x / N[(y * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(-2.0 * N[(a * N[(y * b), $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)))))) < -2e17

   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 -2e17 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 9.99999999999999961e225

   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 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 67.0

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

   5. Simplified 67.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 39.3

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

   8. Simplified 39.3%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{x}{y} + y \cdot 1}} \]

       3. *-rgt-identity N/A

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

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

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

       5. /-lowering-/.f64 48.1

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

   11. Simplified 48.1%

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

   if 9.99999999999999961e225 < (-.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 95.1%

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

   3. Taylor expanded in b around inf

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

      1. *-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 66.1

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

   5. Simplified 66.1%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-lowering-+.f64 54.1

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

   8. Simplified 54.1%

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

   9. Taylor expanded in a around inf

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

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

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

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

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

       3. *-lowering-*.f64 38.9

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

   11. Simplified 38.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+226}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \frac{x}{y}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + -2 \cdot \left(a \cdot \left(y \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+226}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \frac{x}{y}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -0.5}{a \cdot \left(y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 1e+226)
       (/ x (fma y (/ x y) y))
       (/ (* x -0.5) (* a (* y b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 1e+226) {
		tmp = x / fma(y, (x / y), y);
	} else {
		tmp = (x * -0.5) / (a * (y * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 1e+226)
		tmp = Float64(x / fma(y, Float64(x / y), y));
	else
		tmp = Float64(Float64(x * -0.5) / Float64(a * Float64(y * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 1e+226], N[(x / N[(y * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.5), $MachinePrecision] / N[(a * N[(y * b), $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)))))) < -2e17

   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 -2e17 < (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < 9.99999999999999961e225

   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 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 67.0

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

   5. Simplified 67.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 39.3

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

   8. Simplified 39.3%

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

   9. Taylor expanded in y around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

          \[\leadsto \frac{x}{\color{blue}{y \cdot \frac{x}{y} + y \cdot 1}} \]

       3. *-rgt-identity N/A

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

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

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

       5. /-lowering-/.f64 48.1

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

   11. Simplified 48.1%

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

   if 9.99999999999999961e225 < (-.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 95.1%

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

   3. Taylor expanded in b around inf

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

      1. *-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 66.1

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

   5. Simplified 66.1%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-lowering-+.f64 54.1

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

   8. Simplified 54.1%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

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

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

       5. *-lowering-*.f64 37.5

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

   11. Simplified 37.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 10^{+226}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \frac{x}{y}, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -0.5}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -0.5}{a \cdot \left(y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (+
          (/ (* z (sqrt (+ t a))) t)
          (* (- b c) (- (/ 2.0 (* t 3.0)) (+ a (/ 5.0 6.0)))))))
   (if (<= t_1 -2e+17)
     1.0
     (if (<= t_1 2e+167) (/ x (+ x y)) (/ (* x -0.5) (* a (* y b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 2e+167) {
		tmp = x / (x + y);
	} else {
		tmp = (x * -0.5) / (a * (y * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))
    if (t_1 <= (-2d+17)) then
        tmp = 1.0d0
    else if (t_1 <= 2d+167) then
        tmp = x / (x + y)
    else
        tmp = (x * (-0.5d0)) / (a * (y * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	double tmp;
	if (t_1 <= -2e+17) {
		tmp = 1.0;
	} else if (t_1 <= 2e+167) {
		tmp = x / (x + y);
	} else {
		tmp = (x * -0.5) / (a * (y * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = ((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))
	tmp = 0
	if t_1 <= -2e+17:
		tmp = 1.0
	elif t_1 <= 2e+167:
		tmp = x / (x + y)
	else:
		tmp = (x * -0.5) / (a * (y * b))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(a + Float64(5.0 / 6.0)))))
	tmp = 0.0
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 2e+167)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = Float64(Float64(x * -0.5) / Float64(a * Float64(y * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))));
	tmp = 0.0;
	if (t_1 <= -2e+17)
		tmp = 1.0;
	elseif (t_1 <= 2e+167)
		tmp = x / (x + y);
	else
		tmp = (x * -0.5) / (a * (y * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+17], 1.0, If[LessEqual[t$95$1, 2e+167], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.5), $MachinePrecision] / N[(a * N[(y * b), $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)))))) < -2e17

   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 -2e17 < (-.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)))))) < 2.0000000000000001e167

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-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 73.3

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

   5. Simplified 73.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 51.3

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

   8. Simplified 51.3%

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

   if 2.0000000000000001e167 < (-.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 95.8%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 64.1

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

   5. Simplified 64.1%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. +-lowering-+.f64 51.9

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

   8. Simplified 51.9%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

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

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

       5. *-lowering-*.f64 35.3

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

   11. Simplified 35.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq 2 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -0.5}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -330:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \end{array} \end{array} \]

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)))))
      -330.0)
   1.0
   (/ 1.0 (/ (+ x y) 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))))) <= -330.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / ((x + y) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))) <= (-330.0d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / ((x + y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -330.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / ((x + y) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -330.0:
		tmp = 1.0
	else:
		tmp = 1.0 / ((x + y) / 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))))) <= -330.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(Float64(x + y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -330.0)
		tmp = 1.0;
	else
		tmp = 1.0 / ((x + y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -330.0], 1.0, N[(1.0 / N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))) < -330

   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 -330 < (-.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 96.7%

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

   3. Taylor expanded in b around inf

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

      1. *-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 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 22.8

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

   8. Simplified 22.8%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y + x}{x}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{y + x}{x}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{y + x}{x}}} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x + y}}{x}} \]

      5. +-lowering-+.f64 23.2

         \[\leadsto \frac{1}{\frac{\color{blue}{x + y}}{x}} \]

   10. Applied egg-rr 23.2%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -330:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 58.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -330:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \end{array} \]

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)))))
      -330.0)
   1.0
   (/ x (+ x y))))

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))))) <= -330.0) {
		tmp = 1.0;
	} else {
		tmp = x / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - (a + (5.0d0 / 6.0d0))))) <= (-330.0d0)) then
        tmp = 1.0d0
    else
        tmp = x / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -330.0) {
		tmp = 1.0;
	} else {
		tmp = x / (x + y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -330.0:
		tmp = 1.0
	else:
		tmp = x / (x + y)
	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))))) <= -330.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - (a + (5.0 / 6.0))))) <= -330.0)
		tmp = 1.0;
	else
		tmp = x / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -330.0], 1.0, N[(x / N[(x + y), $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)))))) < -330

   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 -330 < (-.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 96.7%

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

   3. Taylor expanded in b around inf

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

      1. *-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 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 22.8

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

   8. Simplified 22.8%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right) \leq -330:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 51.4% 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 97.6%

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

3. Taylor expanded in x around inf

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

5. Simplified 45.2%

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

   1. *-inverses 45.2

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

7. Applied egg-rr 45.2%

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

Developer Target 1: 95.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(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)))))))))))