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

Percentage Accurate: 94.2% → 96.8%

Time: 18.5s

Alternatives: 11

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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)) - ((5.0 / 6.0) + a)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x / (x + (y * Math.exp((2.0 * t_1))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (b * ((0.6666666666666666 / t) - (0.8333333333333334 + a)))))));
	}
	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)) - ((5.0 / 6.0) + a)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = x / (x + (y * math.exp((2.0 * t_1))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (b * ((0.6666666666666666 / t) - (0.8333333333333334 + a)))))))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-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 70.1

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

   5. Simplified 70.1%

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

4. Final simplification 98.1%

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

Alternative 2: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(0.8333333333333334 + a\right) + \frac{-0.6666666666666666}{t}\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(t\_1 \cdot t\_1\right), y \cdot t\_1\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (+ 0.8333333333333334 a) (/ -0.6666666666666666 t))))
   (if (<=
        (/
         x
         (+
          x
          (*
           y
           (exp
            (*
             2.0
             (+
              (/ (* z (sqrt (+ t a))) t)
              (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
        5e-25)
     (/ x (+ x (fma c (* 2.0 (fma c (* y (* t_1 t_1)) (* y t_1))) y)))
     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 tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / (x + fma(c, (2.0 * fma(c, (y * (t_1 * t_1)), (y * t_1))), y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.8333333333333334 + a) + Float64(-0.6666666666666666 / t))
	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(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(x / Float64(x + fma(c, Float64(2.0 * fma(c, Float64(y * Float64(t_1 * t_1)), Float64(y * t_1))), y)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.8333333333333334 + a), $MachinePrecision] + N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / N[(x + N[(c * N[(2.0 * N[(c * N[(y * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 66.6

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

   5. Simplified 66.6%

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

   6. Taylor expanded in c around 0

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

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

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

      2. +-commutative N/A

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

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

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

   8. Simplified 67.9%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, y \cdot \left(\left(\left(0.8333333333333334 + a\right) + \frac{-0.6666666666666666}{t}\right) \cdot \left(\left(0.8333333333333334 + a\right) + \frac{-0.6666666666666666}{t}\right)\right), y \cdot \left(\left(0.8333333333333334 + a\right) + \frac{-0.6666666666666666}{t}\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.6666666666666666}{t} - \left(0.8333333333333334 + a\right)\\ \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(b, 2 \cdot \left(\left(b \cdot y\right) \cdot \left(t\_1 \cdot t\_1\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (/ 0.6666666666666666 t) (+ 0.8333333333333334 a))))
   (if (<=
        (/
         x
         (+
          x
          (*
           y
           (exp
            (*
             2.0
             (+
              (/ (* z (sqrt (+ t a))) t)
              (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
        5e-25)
     (/ x (+ x (fma b (* 2.0 (* (* b y) (* t_1 t_1))) y)))
     1.0)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (0.6666666666666666 / t) - (0.8333333333333334 + a);
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / (x + fma(b, (2.0 * ((b * y) * (t_1 * t_1))), y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(0.6666666666666666 / t) - Float64(0.8333333333333334 + a))
	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(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(x / Float64(x + fma(b, Float64(2.0 * Float64(Float64(b * y) * Float64(t_1 * t_1))), y)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / N[(x + N[(b * N[(2.0 * N[(N[(b * y), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   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 61.5

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

   5. Simplified 61.5%

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

   6. Taylor expanded in b around 0

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

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

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

      2. +-commutative N/A

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

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

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

   8. Simplified 65.0%

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

   9. Taylor expanded in b around inf

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

       1. associate-*r* N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

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

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

       10. +-commutative N/A

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

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

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

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

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

       13. associate-*r/ N/A

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

       14. metadata-eval N/A

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

       15. /-lowering-/.f64 N/A

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

       16. +-commutative N/A

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

       17. +-lowering-+.f64 66.0

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

   11. Simplified 66.0%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x + \mathsf{fma}\left(b, 2 \cdot \left(\left(b \cdot y\right) \cdot \left(\left(\frac{0.6666666666666666}{t} - \left(0.8333333333333334 + a\right)\right) \cdot \left(\frac{0.6666666666666666}{t} - \left(0.8333333333333334 + a\right)\right)\right)\right), y\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + a\right) \cdot \left(0.8333333333333334 + a\right), 0.8333333333333334 + a\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
      5e-25)
   (/
    x
    (fma
     y
     (fma
      c
      (*
       2.0
       (fma
        c
        (* (+ 0.8333333333333334 a) (+ 0.8333333333333334 a))
        (+ 0.8333333333333334 a)))
      1.0)
     x))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / fma(y, fma(c, (2.0 * fma(c, ((0.8333333333333334 + a) * (0.8333333333333334 + a)), (0.8333333333333334 + a))), 1.0), x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(x / fma(y, fma(c, Float64(2.0 * fma(c, Float64(Float64(0.8333333333333334 + a) * Float64(0.8333333333333334 + a)), Float64(0.8333333333333334 + a))), 1.0), x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / N[(y * N[(c * N[(2.0 * N[(c * N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision] + N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 66.6

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

   5. Simplified 66.6%

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

   6. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. +-lowering-+.f64 55.4

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

   8. Simplified 55.4%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. 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. +-commutative N/A

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

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

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

       10. +-commutative N/A

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

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

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

       12. +-commutative N/A

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

       13. +-lowering-+.f64 63.1

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

   11. Simplified 63.1%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(c, 2 \cdot \mathsf{fma}\left(c, \left(0.8333333333333334 + a\right) \cdot \left(0.8333333333333334 + a\right), 0.8333333333333334 + a\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(b, \left(0.8333333333333334 + a\right) \cdot \mathsf{fma}\left(b, 2 \cdot \left(0.8333333333333334 + a\right), -2\right), 1\right), y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
      5e-25)
   (/
    x
    (fma
     (fma
      b
      (*
       (+ 0.8333333333333334 a)
       (fma b (* 2.0 (+ 0.8333333333333334 a)) -2.0))
      1.0)
     y
     x))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / fma(fma(b, ((0.8333333333333334 + a) * fma(b, (2.0 * (0.8333333333333334 + a)), -2.0)), 1.0), y, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(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(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(x / fma(fma(b, Float64(Float64(0.8333333333333334 + a) * fma(b, Float64(2.0 * Float64(0.8333333333333334 + a)), -2.0)), 1.0), y, x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / N[(N[(b * N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b * N[(2.0 * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   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 61.5

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

   5. Simplified 61.5%

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

   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 42.5

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

   8. Simplified 42.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + b \cdot \left(-2 \cdot \left(\frac{5}{6} + a\right) + 2 \cdot \left(b \cdot {\left(\frac{5}{6} + a\right)}^{2}\right)\right)}, x\right)} \]
   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. accelerator-lowering-fma.f64 N/A

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

       4. +-commutative N/A

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

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

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

       6. associate-*r* N/A

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

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

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

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

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

       9. unpow2 N/A

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

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

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

       11. +-commutative N/A

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

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

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

       13. +-commutative N/A

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

       14. +-lowering-+.f64 58.5

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

   11. Simplified 58.5%

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

       1. *-commutative N/A

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

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

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

   13. Applied egg-rr 54.0%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(b, \left(0.8333333333333334 + a\right) \cdot \mathsf{fma}\left(b, 2 \cdot \left(0.8333333333333334 + a\right), -2\right), 1\right), y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
      5e-25)
   (* (- y x) (/ x (* (+ x y) (- y x))))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - ((5.0d0 / 6.0d0) + a))))))))) <= 5d-25) then
        tmp = (y - x) * (x / ((x + y) * (y - x)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25:
		tmp = (y - x) * (x / ((x + y) * (y - x)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(Float64(y - x) * Float64(x / Float64(Float64(x + y) * Float64(y - x))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = (y - x) * (x / ((x + y) * (y - x)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(N[(y - x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   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 61.5

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

   5. Simplified 61.5%

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

   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 19.7

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

   8. Simplified 19.7%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \]
   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 51.9

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

   10. Applied egg-rr 51.9%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{x}{\left(x + y\right) \cdot \left(y - x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 1.3888888888888888, -1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
      5e-25)
   (/ x (fma y (fma b (fma b 1.3888888888888888 -1.6666666666666667) 1.0) x))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / fma(y, fma(b, fma(b, 1.3888888888888888, -1.6666666666666667), 1.0), x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(x / fma(y, fma(b, fma(b, 1.3888888888888888, -1.6666666666666667), 1.0), x));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / N[(y * N[(b * N[(b * 1.3888888888888888 + -1.6666666666666667), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   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 61.5

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

   5. Simplified 61.5%

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

   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 42.5

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

   8. Simplified 42.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{1 + b \cdot \left(-2 \cdot \left(\frac{5}{6} + a\right) + 2 \cdot \left(b \cdot {\left(\frac{5}{6} + a\right)}^{2}\right)\right)}, x\right)} \]
   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. accelerator-lowering-fma.f64 N/A

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

       4. +-commutative N/A

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

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

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

       6. associate-*r* N/A

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

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

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

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

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

       9. unpow2 N/A

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

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

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

       11. +-commutative N/A

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

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

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

       13. +-commutative N/A

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

       14. +-lowering-+.f64 58.5

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

   11. Simplified 58.5%

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

   12. Taylor expanded in a around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. metadata-eval 41.2

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

   14. Simplified 41.2%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 1.3888888888888888, -1.6666666666666667\right), 1\right), x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        x
        (*
         y
         (exp
          (*
           2.0
           (+
            (/ (* z (sqrt (+ t a))) t)
            (* (- b c) (- (/ 2.0 (* t 3.0)) (+ (/ 5.0 6.0) a)))))))))
      5e-25)
   (/ x (+ x y))
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / (x + y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x / (x + (y * exp((2.0d0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0d0 / (t * 3.0d0)) - ((5.0d0 / 6.0d0) + a))))))))) <= 5d-25) then
        tmp = x / (x + y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / (x + (y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		tmp = x / (x + y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25:
		tmp = x / (x + y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) + Float64(Float64(b - c) * Float64(Float64(2.0 / Float64(t * 3.0)) - Float64(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = Float64(x / Float64(x + y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) + ((b - c) * ((2.0 / (t * 3.0)) - ((5.0 / 6.0) + a))))))))) <= 5e-25)
		tmp = x / (x + y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(N[(b - c), $MachinePrecision] * N[(N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   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 61.5

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

   5. Simplified 61.5%

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

   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 19.7

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

   8. Simplified 19.7%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)) (+ (/ 5.0 6.0) a)))))))))
      5e-25)
   (/ 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)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		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)) - ((5.0d0 / 6.0d0) + a))))))))) <= 5d-25) 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)) - ((5.0 / 6.0) + a))))))))) <= 5e-25) {
		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)) - ((5.0 / 6.0) + a))))))))) <= 5e-25:
		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(Float64(5.0 / 6.0) + a))))))))) <= 5e-25)
		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)) - ((5.0 / 6.0) + a))))))))) <= 5e-25)
		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[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-25], N[(x / y), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 4.99999999999999962e-25

   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 61.5

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

   5. Simplified 61.5%

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

   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 19.7

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

   8. Simplified 19.7%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 19.2

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

   11. Simplified 19.2%

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

   if 4.99999999999999962e-25 < (/.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 89.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 95.0%

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

      1. *-inverses 95.0

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

   7. Applied egg-rr 95.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(\frac{5}{6} + a\right)\right)\right)}} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right) + a\right)\right)}}\\ \mathbf{if}\;c \leq -8.5 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 3.5:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(0.8333333333333334 + a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (/
          x
          (+
           x
           (*
            y
            (exp
             (*
              2.0
              (*
               c
               (+ (+ 0.8333333333333334 (/ -0.6666666666666666 t)) a)))))))))
   (if (<= c -8.5e+58)
     t_1
     (if (<= c 3.5)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (* b (- (/ 0.6666666666666666 t) (+ 0.8333333333333334 a))))))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x / (x + (y * exp((2.0 * (c * ((0.8333333333333334 + (-0.6666666666666666 / t)) + a))))));
	double tmp;
	if (c <= -8.5e+58) {
		tmp = t_1;
	} else if (c <= 3.5) {
		tmp = x / (x + (y * exp((2.0 * (b * ((0.6666666666666666 / t) - (0.8333333333333334 + a)))))));
	} else {
		tmp = t_1;
	}
	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 = x / (x + (y * exp((2.0d0 * (c * ((0.8333333333333334d0 + ((-0.6666666666666666d0) / t)) + a))))))
    if (c <= (-8.5d+58)) then
        tmp = t_1
    else if (c <= 3.5d0) then
        tmp = x / (x + (y * exp((2.0d0 * (b * ((0.6666666666666666d0 / t) - (0.8333333333333334d0 + a)))))))
    else
        tmp = t_1
    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 = x / (x + (y * Math.exp((2.0 * (c * ((0.8333333333333334 + (-0.6666666666666666 / t)) + a))))));
	double tmp;
	if (c <= -8.5e+58) {
		tmp = t_1;
	} else if (c <= 3.5) {
		tmp = x / (x + (y * Math.exp((2.0 * (b * ((0.6666666666666666 / t) - (0.8333333333333334 + a)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = x / (x + (y * math.exp((2.0 * (c * ((0.8333333333333334 + (-0.6666666666666666 / t)) + a))))))
	tmp = 0
	if c <= -8.5e+58:
		tmp = t_1
	elif c <= 3.5:
		tmp = x / (x + (y * math.exp((2.0 * (b * ((0.6666666666666666 / t) - (0.8333333333333334 + a)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * Float64(Float64(0.8333333333333334 + Float64(-0.6666666666666666 / t)) + a)))))))
	tmp = 0.0
	if (c <= -8.5e+58)
		tmp = t_1;
	elseif (c <= 3.5)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(b * Float64(Float64(0.6666666666666666 / t) - Float64(0.8333333333333334 + a))))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = x / (x + (y * exp((2.0 * (c * ((0.8333333333333334 + (-0.6666666666666666 / t)) + a))))));
	tmp = 0.0;
	if (c <= -8.5e+58)
		tmp = t_1;
	elseif (c <= 3.5)
		tmp = x / (x + (y * exp((2.0 * (b * ((0.6666666666666666 / t) - (0.8333333333333334 + a)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * N[(N[(0.8333333333333334 + N[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8.5e+58], t$95$1, If[LessEqual[c, 3.5], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(b * N[(N[(0.6666666666666666 / t), $MachinePrecision] - N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -8.50000000000000015e58 or 3.5 < c

   1. Initial program 92.5%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 88.7

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

   5. Simplified 88.7%

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

   if -8.50000000000000015e58 < c < 3.5

   1. Initial program 96.3%

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

   3. Taylor expanded in b around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 78.9

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

   5. Simplified 78.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right) + a\right)\right)}}\\ \mathbf{elif}\;c \leq 3.5:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(b \cdot \left(\frac{0.6666666666666666}{t} - \left(0.8333333333333334 + a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(\left(0.8333333333333334 + \frac{-0.6666666666666666}{t}\right) + a\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.3% accurate, 198.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 49.5%

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

   1. *-inverses 49.5

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

7. Applied egg-rr 49.5%

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

Developer Target 1: 95.5% 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 2024205 
(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)))))))))))