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

Percentage Accurate: 93.7% → 96.5%

Time: 16.1s

Alternatives: 7

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: 93.7% 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: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 64.3%

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

      1. lift-+.f64 N/A

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

   10. Applied rewrites 64.3%

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

4. Final simplification 97.0%

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

Alternative 2: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\frac{2}{3 \cdot t} - \left(\frac{5}{6} + a\right)\right) \cdot \left(c - b\right)\right) \cdot 2} \cdot y + x} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        (*
         (exp
          (*
           (-
            (/ (* (sqrt (+ a t)) z) t)
            (* (- (/ 2.0 (* 3.0 t)) (+ (/ 5.0 6.0) a)) (- c b)))
           2.0))
         y)
        x))
      5e-32)
   (/
    x
    (+
     (*
      (exp (* (* (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)) c) 2.0))
      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 / ((exp(((((sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 5e-32) {
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * 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 / ((exp(((((sqrt((a + t)) * z) / t) - (((2.0d0 / (3.0d0 * t)) - ((5.0d0 / 6.0d0) + a)) * (c - b))) * 2.0d0)) * y) + x)) <= 5d-32) then
        tmp = x / ((exp(((((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t)) * c) * 2.0d0)) * 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 / ((Math.exp(((((Math.sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 5e-32) {
		tmp = x / ((Math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / ((math.exp(((((math.sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 5e-32:
		tmp = x / ((math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * 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(Float64(exp(Float64(Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(2.0 / Float64(3.0 * t)) - Float64(Float64(5.0 / 6.0) + a)) * Float64(c - b))) * 2.0)) * y) + x)) <= 5e-32)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)) * c) * 2.0)) * 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 / ((exp(((((sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 5e-32)
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * 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[(N[(N[Exp[N[(N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 5e-32], N[(x / N[(N[(N[Exp[N[(N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $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))))))))))) < 5e-32

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if 5e-32 < (/.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 85.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 93.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\frac{2}{3 \cdot t} - \left(\frac{5}{6} + a\right)\right) \cdot \left(c - b\right)\right) \cdot 2} \cdot y + x} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\frac{2}{3 \cdot t} - \left(\frac{5}{6} + a\right)\right) \cdot \left(c - b\right)\right) \cdot 2} \cdot y + x} \leq 0:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.5} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (/
       x
       (+
        (*
         (exp
          (*
           (-
            (/ (* (sqrt (+ a t)) z) t)
            (* (- (/ 2.0 (* 3.0 t)) (+ (/ 5.0 6.0) a)) (- c b)))
           2.0))
         y)
        x))
      0.0)
   (* (pow (* x x) -0.5) x)
   1.0))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x / ((exp(((((sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 0.0) {
		tmp = pow((x * x), -0.5) * 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 / ((exp(((((sqrt((a + t)) * z) / t) - (((2.0d0 / (3.0d0 * t)) - ((5.0d0 / 6.0d0) + a)) * (c - b))) * 2.0d0)) * y) + x)) <= 0.0d0) then
        tmp = ((x * x) ** (-0.5d0)) * 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 / ((Math.exp(((((Math.sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 0.0) {
		tmp = Math.pow((x * x), -0.5) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x / ((math.exp(((((math.sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 0.0:
		tmp = math.pow((x * x), -0.5) * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(2.0 / Float64(3.0 * t)) - Float64(Float64(5.0 / 6.0) + a)) * Float64(c - b))) * 2.0)) * y) + x)) <= 0.0)
		tmp = Float64((Float64(x * x) ^ -0.5) * 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 / ((exp(((((sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) * 2.0)) * y) + x)) <= 0.0)
		tmp = ((x * x) ^ -0.5) * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x / N[(N[(N[Exp[N[(N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[N[(x * x), $MachinePrecision], -0.5], $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot x \]
   9. Step-by-step derivation

      1. lower-/.f64 3.1

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

   10. Applied rewrites 3.1%

       \[\leadsto \color{blue}{\frac{1}{x}} \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 26.0%

       \[\leadsto {\left(x \cdot x\right)}^{\color{blue}{-0.5}} \cdot x \]

   if -0.0 < (/.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 85.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{e^{\left(\frac{\sqrt{a + t} \cdot z}{t} - \left(\frac{2}{3 \cdot t} - \left(\frac{5}{6} + a\right)\right) \cdot \left(c - b\right)\right) \cdot 2} \cdot y + x} \leq 0:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.5} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[Exp[N[(N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], 0.0], 1.0, N[(x / N[(N[Exp[N[(N[(c * a), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))) < 0.0

   1. Initial program 99.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.1%

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

   if 0.0 < (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))

   1. Initial program 87.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   7. Applied rewrites 69.8%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 55.0%

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

   12. Applied rewrites 55.1%

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

4. Final simplification 73.3%

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

Alternative 5: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\frac{2}{3 \cdot t} - \left(\frac{5}{6} + a\right)\right) \cdot \left(c - b\right) \leq -1000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (-
       (/ (* (sqrt (+ a t)) z) t)
       (* (- (/ 2.0 (* 3.0 t)) (+ (/ 5.0 6.0) a)) (- c b)))
      -1000000.0)
   1.0
   (/ x (+ (* (exp (* (* (+ 0.8333333333333334 a) c) 2.0)) y) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) <= -1000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((((sqrt((a + t)) * z) / t) - (((2.0d0 / (3.0d0 * t)) - ((5.0d0 / 6.0d0) + a)) * (c - b))) <= (-1000000.0d0)) then
        tmp = 1.0d0
    else
        tmp = x / ((exp((((0.8333333333333334d0 + a) * c) * 2.0d0)) * y) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((((Math.sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) <= -1000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / ((Math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (((math.sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) <= -1000000.0:
		tmp = 1.0
	else:
		tmp = x / ((math.exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(sqrt(Float64(a + t)) * z) / t) - Float64(Float64(Float64(2.0 / Float64(3.0 * t)) - Float64(Float64(5.0 / 6.0) + a)) * Float64(c - b))) <= -1000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(0.8333333333333334 + a) * c) * 2.0)) * y) + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((((sqrt((a + t)) * z) / t) - (((2.0 / (3.0 * t)) - ((5.0 / 6.0) + a)) * (c - b))) <= -1000000.0)
		tmp = 1.0;
	else
		tmp = x / ((exp((((0.8333333333333334 + a) * c) * 2.0)) * y) + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision] - N[(N[(N[(2.0 / N[(3.0 * t), $MachinePrecision]), $MachinePrecision] - N[(N[(5.0 / 6.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1000000.0], 1.0, N[(x / N[(N[(N[Exp[N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.1%

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

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

   1. Initial program 87.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 64.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{a + t} \cdot z}{t} - \left(\frac{2}{3 \cdot t} - \left(\frac{5}{6} + a\right)\right) \cdot \left(c - b\right) \leq -1000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(0.8333333333333334 + a\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{e^{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (/
          x
          (+
           (*
            (exp
             (*
              (* (- (- (/ 0.6666666666666666 t) 0.8333333333333334) a) b)
              2.0))
            y)
           x))))
   (if (<= b -5.5e+42)
     t_1
     (if (<= b 5.3e+113)
       (/
        x
        (+
         (*
          (exp
           (* (* (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)) c) 2.0))
          y)
         x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x / ((exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x);
	double tmp;
	if (b <= -5.5e+42) {
		tmp = t_1;
	} else if (b <= 5.3e+113) {
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	} 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 / ((exp((((((0.6666666666666666d0 / t) - 0.8333333333333334d0) - a) * b) * 2.0d0)) * y) + x)
    if (b <= (-5.5d+42)) then
        tmp = t_1
    else if (b <= 5.3d+113) then
        tmp = x / ((exp(((((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t)) * c) * 2.0d0)) * y) + x)
    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 / ((Math.exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x);
	double tmp;
	if (b <= -5.5e+42) {
		tmp = t_1;
	} else if (b <= 5.3e+113) {
		tmp = x / ((Math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = x / ((math.exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x)
	tmp = 0
	if b <= -5.5e+42:
		tmp = t_1
	elif b <= 5.3e+113:
		tmp = x / ((math.exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(Float64(0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x))
	tmp = 0.0
	if (b <= -5.5e+42)
		tmp = t_1;
	elseif (b <= 5.3e+113)
		tmp = Float64(x / Float64(Float64(exp(Float64(Float64(Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)) * c) * 2.0)) * y) + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = x / ((exp((((((0.6666666666666666 / t) - 0.8333333333333334) - a) * b) * 2.0)) * y) + x);
	tmp = 0.0;
	if (b <= -5.5e+42)
		tmp = t_1;
	elseif (b <= 5.3e+113)
		tmp = x / ((exp(((((0.8333333333333334 + a) - (0.6666666666666666 / t)) * c) * 2.0)) * y) + x);
	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[(N[(N[Exp[N[(N[(N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - 0.8333333333333334), $MachinePrecision] - a), $MachinePrecision] * b), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.5e+42], t$95$1, If[LessEqual[b, 5.3e+113], N[(x / N[(N[(N[Exp[N[(N[(N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.50000000000000001e42 or 5.29999999999999967e113 < b

   1. Initial program 87.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -5.50000000000000001e42 < b < 5.29999999999999967e113

   1. Initial program 95.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+113}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right) \cdot c\right) \cdot 2} \cdot y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\left(\left(\frac{0.6666666666666666}{t} - 0.8333333333333334\right) - a\right) \cdot b\right) \cdot 2} \cdot y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% 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 92.2%

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

3. Taylor expanded in c around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 69.2

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

5. Applied rewrites 69.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 50.9%

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

Developer Target 1: 95.4% 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 2024235 
(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)))))))))))