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

Percentage Accurate: 93.5% → 97.5%

Time: 11.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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: 93.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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: 97.5% accurate, 0.3× speedup

Math

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

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) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))
      2.0)
   (/
    x
    (*
     x
     (+
      1.0
      (/
       (*
        y
        (pow
         (exp 2.0)
         (-
          (* (/ z t) (sqrt (+ a t)))
          (*
           (- b c)
           (- (+ 0.8333333333333334 a) (* 0.6666666666666666 (pow t -1.0)))))))
       x))))
   (/
    x
    (+ x (* y (exp (* 2.0 (* a (- (+ c (* (/ 1.0 (sqrt a)) (/ z t))) b)))))))))

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))) <= 2.0d0) then
        tmp = x / (x * (1.0d0 + ((y * (exp(2.0d0) ** (((z / t) * sqrt((a + t))) - ((b - c) * ((0.8333333333333334d0 + a) - (0.6666666666666666d0 * (t ** (-1.0d0)))))))) / x)))
    else
        tmp = x / (x + (y * exp((2.0d0 * (a * ((c + ((1.0d0 / sqrt(a)) * (z / t))) - b))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((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)))))))))) <= 2.0) {
		tmp = x / (x * (1.0 + ((y * Math.pow(Math.exp(2.0), (((z / t) * Math.sqrt((a + t))) - ((b - c) * ((0.8333333333333334 + a) - (0.6666666666666666 * Math.pow(t, -1.0))))))) / x)));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * (a * ((c + ((1.0 / Math.sqrt(a)) * (z / t))) - b))))));
	}
	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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 2.0:
		tmp = x / (x * (1.0 + ((y * math.pow(math.exp(2.0), (((z / t) * math.sqrt((a + t))) - ((b - c) * ((0.8333333333333334 + a) - (0.6666666666666666 * math.pow(t, -1.0))))))) / x)))
	else:
		tmp = x / (x + (y * math.exp((2.0 * (a * ((c + ((1.0 / math.sqrt(a)) * (z / t))) - b))))))
	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(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0)))))))))) <= 2.0)
		tmp = Float64(x / Float64(x * Float64(1.0 + Float64(Float64(y * (exp(2.0) ^ Float64(Float64(Float64(z / t) * sqrt(Float64(a + t))) - Float64(Float64(b - c) * Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 * (t ^ -1.0))))))) / x))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(Float64(c + Float64(Float64(1.0 / sqrt(a)) * Float64(z / t))) - b)))))));
	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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 2.0)
		tmp = x / (x * (1.0 + ((y * (exp(2.0) ^ (((z / t) * sqrt((a + t))) - ((b - c) * ((0.8333333333333334 + a) - (0.6666666666666666 * (t ^ -1.0))))))) / x)));
	else
		tmp = x / (x + (y * exp((2.0 * (a * ((c + ((1.0 / sqrt(a)) * (z / t))) - b))))));
	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[(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], 2.0], N[(x / N[(x * N[(1.0 + N[(N[(y * N[Power[N[Exp[2.0], $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 * N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(N[(c + N[(N[(1.0 / N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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))))))))))) < 2

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

   if 2 < (/.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 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 a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(a \cdot \left(\left(c + \frac{1}{\sqrt{a}} \cdot \frac{z}{t}\right) - b\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.9% accurate, 0.5× speedup

Math

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

FPCore

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

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(x, y, z, t, a, b, c):
	t_1 = 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)))))))))
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_1
	else:
		tmp = x / (x + (y * math.exp((2.0 * (a * ((c + ((1.0 / math.sqrt(a)) * (z / t))) - b))))))
	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(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))))))))))
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(Float64(c + Float64(Float64(1.0 / sqrt(a)) * Float64(z / t))) - b)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	tmp = 0.0;
	if (t_1 <= 2.0)
		tmp = t_1;
	else
		tmp = x / (x + (y * exp((2.0 * (a * ((c + ((1.0 / sqrt(a)) * (z / t))) - b))))));
	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[(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]}, If[LessEqual[t$95$1, 2.0], t$95$1, N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(N[(c + N[(N[(1.0 / N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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))))))))))) < 2

   1. Initial program 98.8%

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

   if 2 < (/.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 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 a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(a \cdot \left(\left(c + \frac{1}{\sqrt{a}} \cdot \frac{z}{t}\right) - b\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \leq 0.05:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))
      0.05)
   (/ x (+ x (* y (exp (* 2.0 (* c (+ 0.8333333333333334 a)))))))
   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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05) {
		tmp = x / (x + (y * exp((2.0 * (c * (0.8333333333333334 + a))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))) <= 0.05d0) then
        tmp = x / (x + (y * exp((2.0d0 * (c * (0.8333333333333334d0 + a))))))
    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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05) {
		tmp = x / (x + (y * Math.exp((2.0 * (c * (0.8333333333333334 + a))))));
	} 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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05:
		tmp = x / (x + (y * math.exp((2.0 * (c * (0.8333333333333334 + a))))))
	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(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0)))))))))) <= 0.05)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * Float64(0.8333333333333334 + a)))))));
	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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05)
		tmp = x / (x + (y * exp((2.0 * (c * (0.8333333333333334 + a))))));
	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[(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], 0.05], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 0.050000000000000003

   1. Initial program 98.3%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in c 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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. metadata-eval 56.4

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

   8. Applied rewrites 56.4%

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

   if 0.050000000000000003 < (/.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 88.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.8%

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

Alternative 4: 70.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \leq 0.05:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot 0.8333333333333334\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))
      0.05)
   (/ x (+ x (* y (exp (* 2.0 (* c 0.8333333333333334))))))
   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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05) {
		tmp = x / (x + (y * exp((2.0 * (c * 0.8333333333333334)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))))) <= 0.05d0) then
        tmp = x / (x + (y * exp((2.0d0 * (c * 0.8333333333333334d0)))))
    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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05) {
		tmp = x / (x + (y * Math.exp((2.0 * (c * 0.8333333333333334)))));
	} 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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05:
		tmp = x / (x + (y * math.exp((2.0 * (c * 0.8333333333333334)))))
	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(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0)))))))))) <= 0.05)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * 0.8333333333333334))))));
	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) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))) <= 0.05)
		tmp = x / (x + (y * exp((2.0 * (c * 0.8333333333333334)))));
	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[(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], 0.05], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 x (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))))) < 0.050000000000000003

   1. Initial program 98.3%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in c 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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. metadata-eval 56.4

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

   8. Applied rewrites 56.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 49.0%

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot 0.8333333333333334\right)}} \]

   if 0.050000000000000003 < (/.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 88.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.8%

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

Alternative 5: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          2.0
          (-
           (/ (* z (sqrt (+ t a))) t)
           (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))
   (if (<= t_1 -1e+18)
     1.0
     (if (<= t_1 5e+307)
       (/ x (+ x (* y (exp (* 2.0 (* c (+ 0.8333333333333334 a)))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = 1.0;
	} else if (t_1 <= 5e+307) {
		tmp = x / (x + (y * exp((2.0 * (c * (0.8333333333333334 + a))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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))))))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = 1.0;
	elseif (t_1 <= 5e+307)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * Float64(0.8333333333333334 + a)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+18], 1.0, If[LessEqual[t$95$1, 5e+307], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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))))))) < -1e18

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

   if -1e18 < (*.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))))))) < 5e307

   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 t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in c 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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. metadata-eval 65.0

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

   8. Applied rewrites 65.0%

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

   if 5e307 < (*.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 75.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 65.2

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

   5. Applied rewrites 65.2%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(0.8333333333333334 + a\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{t} \cdot \sqrt{a + t}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          2.0
          (-
           (/ (* z (sqrt (+ t a))) t)
           (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))
   (if (<= t_1 -1e+18)
     1.0
     (if (<= t_1 5e+307)
       (/ x (+ x (* y (exp (* 2.0 (* c (+ 0.8333333333333334 a)))))))
       (/ x (+ x (* y (exp (* 2.0 (* (/ z t) (sqrt (+ a t))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = 1.0;
	} else if (t_1 <= 5e+307) {
		tmp = x / (x + (y * exp((2.0 * (c * (0.8333333333333334 + a))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * ((z / t) * sqrt((a + t)))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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 = 2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))
    if (t_1 <= (-1d+18)) then
        tmp = 1.0d0
    else if (t_1 <= 5d+307) then
        tmp = x / (x + (y * exp((2.0d0 * (c * (0.8333333333333334d0 + a))))))
    else
        tmp = x / (x + (y * exp((2.0d0 * ((z / t) * sqrt((a + t)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))));
	double tmp;
	if (t_1 <= -1e+18) {
		tmp = 1.0;
	} else if (t_1 <= 5e+307) {
		tmp = x / (x + (y * Math.exp((2.0 * (c * (0.8333333333333334 + a))))));
	} else {
		tmp = x / (x + (y * Math.exp((2.0 * ((z / t) * Math.sqrt((a + t)))))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))
	tmp = 0
	if t_1 <= -1e+18:
		tmp = 1.0
	elif t_1 <= 5e+307:
		tmp = x / (x + (y * math.exp((2.0 * (c * (0.8333333333333334 + a))))))
	else:
		tmp = x / (x + (y * math.exp((2.0 * ((z / t) * math.sqrt((a + t)))))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = 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))))))
	tmp = 0.0
	if (t_1 <= -1e+18)
		tmp = 1.0;
	elseif (t_1 <= 5e+307)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * Float64(0.8333333333333334 + a)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(z / t) * sqrt(Float64(a + t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = 2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))));
	tmp = 0.0;
	if (t_1 <= -1e+18)
		tmp = 1.0;
	elseif (t_1 <= 5e+307)
		tmp = x / (x + (y * exp((2.0 * (c * (0.8333333333333334 + a))))));
	else
		tmp = x / (x + (y * exp((2.0 * ((z / t) * sqrt((a + t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -1e+18], 1.0, If[LessEqual[t$95$1, 5e+307], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(z / t), $MachinePrecision] * N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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))))))) < -1e18

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

   if -1e18 < (*.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))))))) < 5e307

   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 t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in c 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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. metadata-eval 65.0

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

   8. Applied rewrites 65.0%

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

   if 5e307 < (*.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 75.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 61.4

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

   5. Applied rewrites 61.4%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\frac{z}{t} \cdot \sqrt{a + t}\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], 1.0, N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 99.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

   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 87.7%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 57.4

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

   8. Applied rewrites 57.4%

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

Alternative 8: 71.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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 (exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0))))))) <= 0.0d0) then
        tmp = 1.0d0
    else
        tmp = x / (x + (y * exp((2.0d0 * ((-0.8333333333333334d0) * b)))))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0))))))) <= 0.0:
		tmp = 1.0
	else:
		tmp = x / (x + (y * math.exp((2.0 * (-0.8333333333333334 * b)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], 1.0, N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(-0.8333333333333334 * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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 99.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.2%

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

   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 87.7%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. metadata-eval 48.3

         \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(-1 \cdot \left(b \cdot \left(0.8333333333333334 + a\right)\right)\right)}} \]

   8. Applied rewrites 48.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 44.2

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

   11. Applied rewrites 44.2%

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(-0.8333333333333334 \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(\left(c + \frac{1}{\sqrt{a}} \cdot \frac{z}{t}\right) - b\right)\right)}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-283}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -1e-87)
   (/
    x
    (+ x (* y (exp (* 2.0 (* a (- (+ c (* (/ 1.0 (sqrt a)) (/ z t))) b)))))))
   (if (<= t 3.9e-283)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
     (if (<= t 2e-7)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (/ (* z (sqrt (+ t a))) t)
             (* (- b c) (/ -0.6666666666666666 t))))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (* (/ 1.0 (sqrt t)) z)
             (* (+ 0.8333333333333334 a) (- b c))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -1e-87) {
		tmp = x / (x + (y * exp((2.0 * (a * ((c + ((1.0 / sqrt(a)) * (z / t))) - b))))));
	} else if (t <= 3.9e-283) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else if (t <= 2e-7) {
		tmp = x / (x + (y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * (-0.6666666666666666 / t)))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((1.0 / sqrt(t)) * z) - ((0.8333333333333334 + a) * (b - c)))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -1e-87)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(Float64(c + Float64(Float64(1.0 / sqrt(a)) * Float64(z / t))) - b)))))));
	elseif (t <= 3.9e-283)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	elseif (t <= 2e-7)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(-0.6666666666666666 / t))))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(1.0 / sqrt(t)) * z) - Float64(Float64(0.8333333333333334 + a) * Float64(b - c))))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -1e-87], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(N[(c + N[(N[(1.0 / N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.9e-283], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e-7], 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[(-0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.00000000000000002e-87

   1. Initial program 80.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000002e-87 < t < 3.9000000000000002e-283

   1. Initial program 84.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 3.9000000000000002e-283 < t < 1.9999999999999999e-7

   1. Initial program 95.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 1.9999999999999999e-7 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(\left(c + \frac{1}{\sqrt{a}} \cdot \frac{z}{t}\right) - b\right)\right)}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t}}{t} - \left(b - c\right) \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -1e-87)
   (/
    x
    (+ x (* y (exp (* 2.0 (* a (- (+ c (* (/ 1.0 (sqrt a)) (/ z t))) b)))))))
   (if (<= t 6e-185)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
     (if (<= t 4.2e-27)
       (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt t)) t) (* (- b c) a)))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (* (/ 1.0 (sqrt t)) z)
             (* (+ 0.8333333333333334 a) (- b c))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -1e-87) {
		tmp = x / (x + (y * exp((2.0 * (a * ((c + ((1.0 / sqrt(a)) * (z / t))) - b))))));
	} else if (t <= 6e-185) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else if (t <= 4.2e-27) {
		tmp = x / (x + (y * exp((2.0 * (((z * sqrt(t)) / t) - ((b - c) * a))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((1.0 / sqrt(t)) * z) - ((0.8333333333333334 + a) * (b - c)))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -1e-87)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(Float64(c + Float64(Float64(1.0 / sqrt(a)) * Float64(z / t))) - b)))))));
	elseif (t <= 6e-185)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	elseif (t <= 4.2e-27)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(t)) / t) - Float64(Float64(b - c) * a)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(1.0 / sqrt(t)) * z) - Float64(Float64(0.8333333333333334 + a) * Float64(b - c))))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -1e-87], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(N[(c + N[(N[(1.0 / N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-185], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-27], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.00000000000000002e-87

   1. Initial program 80.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000002e-87 < t < 6.00000000000000061e-185

   1. Initial program 87.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 6.00000000000000061e-185 < t < 4.20000000000000031e-27

   1. Initial program 97.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.8%

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

   if 4.20000000000000031e-27 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t}}{t} - \left(b - c\right) \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -2e-42)
   (/ x (+ x (* y (exp (* 2.0 (* a (- c b)))))))
   (if (<= t 6e-185)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
     (if (<= t 4.2e-27)
       (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt t)) t) (* (- b c) a)))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (* (/ 1.0 (sqrt t)) z)
             (* (+ 0.8333333333333334 a) (- b c))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -2e-42) {
		tmp = x / (x + (y * exp((2.0 * (a * (c - b))))));
	} else if (t <= 6e-185) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else if (t <= 4.2e-27) {
		tmp = x / (x + (y * exp((2.0 * (((z * sqrt(t)) / t) - ((b - c) * a))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((1.0 / sqrt(t)) * z) - ((0.8333333333333334 + a) * (b - c)))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -2e-42)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(c - b)))))));
	elseif (t <= 6e-185)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	elseif (t <= 4.2e-27)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(t)) / t) - Float64(Float64(b - c) * a)))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(1.0 / sqrt(t)) * z) - Float64(Float64(0.8333333333333334 + a) * Float64(b - c))))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -2e-42], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-185], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-27], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(1.0 / N[Sqrt[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - N[(N[(0.8333333333333334 + a), $MachinePrecision] * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.00000000000000008e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if -2.00000000000000008e-42 < t < 6.00000000000000061e-185

   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 t around 0

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if 6.00000000000000061e-185 < t < 4.20000000000000031e-27

   1. Initial program 97.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 75.8%

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

   if 4.20000000000000031e-27 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t}}{t} - \left(b - c\right) \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -2e-42)
   (/ x (+ x (* y (exp (* 2.0 (* a (- c b)))))))
   (if (<= t 6e-185)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
     (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt t)) t) (* (- b c) a))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -2e-42) {
		tmp = x / (x + (y * exp((2.0 * (a * (c - b))))));
	} else if (t <= 6e-185) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else {
		tmp = x / (x + (y * exp((2.0 * (((z * sqrt(t)) / t) - ((b - c) * a))))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -2e-42)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(c - b)))))));
	elseif (t <= 6e-185)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(fma(sqrt(a), z, Float64(0.6666666666666666 * Float64(b - c))) / t))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(t)) / t) - Float64(Float64(b - c) * a)))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -2e-42], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(a * N[(c - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-185], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[Sqrt[a], $MachinePrecision] * z + N[(0.6666666666666666 * N[(b - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.00000000000000008e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift--.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if -2.00000000000000008e-42 < t < 6.00000000000000061e-185

   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 t around 0

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if 6.00000000000000061e-185 < t

   1. Initial program 98.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 80.5%

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{\color{blue}{t}}}{t} - \left(b - c\right) \cdot a\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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 93.0%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 53.9%

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

Developer Target 1: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \sqrt{t + a}\\ t_2 := a - \frac{5}{6}\\ \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{t\_1 \cdot \left(\left(3 \cdot t\right) \cdot t\_2\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(t\_2 \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot t\_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{t\_1}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* z (sqrt (+ t a)))) (t_2 (- a (/ 5.0 6.0))))
   (if (< t -2.118326644891581e-50)
     (/
      x
      (+
       x
       (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b)))))))
     (if (< t 5.196588770651547e-123)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (/
             (-
              (* t_1 (* (* 3.0 t) t_2))
              (*
               (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0)
               (* t_2 (* (- b c) t))))
             (* (* (* t t) 3.0) t_2)))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            2.0
            (-
             (/ t_1 t)
             (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = z * sqrt((t + a));
	double t_2 = a - (5.0 / 6.0);
	double tmp;
	if (t < -2.118326644891581e-50) {
		tmp = x / (x + (y * exp((2.0 * (((a * c) + (0.8333333333333334 * c)) - (a * b))))));
	} else if (t < 5.196588770651547e-123) {
		tmp = x / (x + (y * exp((2.0 * (((t_1 * ((3.0 * t) * t_2)) - (((((5.0 / 6.0) + a) * (3.0 * t)) - 2.0) * (t_2 * ((b - c) * t)))) / (((t * t) * 3.0) * t_2))))));
	} else {
		tmp = x / (x + (y * exp((2.0 * ((t_1 / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    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 2025037 
(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)))))))))))