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

Percentage Accurate: 94.2% → 97.5%

Time: 15.0s

Alternatives: 9

Speedup: 0.5×

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: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (/
          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 (/ (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 = 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 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	}
	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(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[(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[(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 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 98.9%

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

   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 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 68.3

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

   5. Applied rewrites 68.3%

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

Alternative 2: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \leq 10^{-63}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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))))))))))
      1e-63)
   (/ x (+ x (* y (exp (* 2.0 (* c 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)))))))))) <= 1e-63) {
		tmp = x / (x + (y * exp((2.0 * (c * 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)))))))))) <= 1d-63) then
        tmp = x / (x + (y * exp((2.0d0 * (c * 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)))))))))) <= 1e-63) {
		tmp = x / (x + (y * Math.exp((2.0 * (c * 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)))))))))) <= 1e-63:
		tmp = x / (x + (y * math.exp((2.0 * (c * 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)))))))))) <= 1e-63)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(c * 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)))))))))) <= 1e-63)
		tmp = x / (x + (y * exp((2.0 * (c * 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], 1e-63], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(c * a), $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))))))))))) < 1.00000000000000007e-63

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 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 59.9

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

   5. Applied rewrites 59.9%

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

   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 52.7

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

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 45.1%

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

   if 1.00000000000000007e-63 < (/.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 90.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.8%

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

Alternative 3: 69.8% accurate, 0.6× speedup

Math

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

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))))))))))
      1e-63)
   (/ x (+ x (* y (exp (* 2.0 (* -0.8333333333333334 b))))))
   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)))))))))) <= 1e-63) {
		tmp = x / (x + (y * exp((2.0 * (-0.8333333333333334 * b)))));
	} 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)))))))))) <= 1d-63) then
        tmp = x / (x + (y * exp((2.0d0 * ((-0.8333333333333334d0) * b)))))
    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)))))))))) <= 1e-63) {
		tmp = x / (x + (y * Math.exp((2.0 * (-0.8333333333333334 * b)))));
	} 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)))))))))) <= 1e-63:
		tmp = x / (x + (y * math.exp((2.0 * (-0.8333333333333334 * b)))))
	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)))))))))) <= 1e-63)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(-0.8333333333333334 * b))))));
	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)))))))))) <= 1e-63)
		tmp = x / (x + (y * exp((2.0 * (-0.8333333333333334 * b)))));
	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], 1e-63], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(-0.8333333333333334 * b), $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))))))))))) < 1.00000000000000007e-63

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 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 59.9

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

   5. Applied rewrites 59.9%

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

   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 51.2

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

   8. Applied rewrites 51.2%

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

   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 42.6

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

   11. Applied rewrites 42.6%

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

   if 1.00000000000000007e-63 < (/.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 90.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.8%

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

Alternative 4: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 10^{+273}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(-1 \cdot \left(b \cdot \left(0.8333333333333334 + a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \end{array} \end{array} \]

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 -5e+33)
     1.0
     (if (<= t_1 1e+273)
       (/ x (+ x (* y (exp (* 2.0 (* -1.0 (* b (+ 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 <= -5e+33) {
		tmp = 1.0;
	} else if (t_1 <= 1e+273) {
		tmp = x / (x + (y * exp((2.0 * (-1.0 * (b * (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 <= -5e+33)
		tmp = 1.0;
	elseif (t_1 <= 1e+273)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(-1.0 * Float64(b * 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, -5e+33], 1.0, If[LessEqual[t$95$1, 1e+273], N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(-1.0 * N[(b * N[(0.8333333333333334 + a), $MachinePrecision]), $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))))))) < -4.99999999999999973e33

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

   if -4.99999999999999973e33 < (*.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))))))) < 9.99999999999999945e272

   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 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in 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 59.4

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

   8. Applied rewrites 59.4%

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

   if 9.99999999999999945e272 < (*.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 81.4%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 63.6

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

   5. Applied rewrites 63.6%

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

Alternative 5: 77.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<=
      (*
       2.0
       (-
        (/ (* z (sqrt (+ t a))) t)
        (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))
      -1e+18)
   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 ((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))) <= -1e+18) {
		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 ((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))) <= (-1d+18)) 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 ((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))) <= -1e+18) {
		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 (2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))) <= -1e+18:
		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 (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)))))) <= -1e+18)
		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 ((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))) <= -1e+18)
		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[(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], -1e+18], 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 (*.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 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

   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)))))))

   1. Initial program 90.4%

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

   3. Taylor expanded in t around 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 60.5

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

   5. Applied rewrites 60.5%

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

   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 61.3

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

   8. Applied rewrites 61.3%

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

Alternative 6: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(\left(c + \frac{1}{\sqrt{a}} \cdot \frac{z}{t}\right) - b\right)\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \frac{-0.6666666666666666}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -5e-76)
   (/
    x
    (+ x (* y (exp (* 2.0 (* a (- (+ c (* (/ 1.0 (sqrt a)) (/ z t))) b)))))))
   (if (<= t 4e-290)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) t))))))
     (if (<= t 1.2e-17)
       (/
        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 <= -5e-76) {
		tmp = x / (x + (y * exp((2.0 * (a * ((c + ((1.0 / sqrt(a)) * (z / t))) - b))))));
	} else if (t <= 4e-290) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / t)))));
	} else if (t <= 1.2e-17) {
		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 <= -5e-76)
		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 <= 4e-290)
		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 <= 1.2e-17)
		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, -5e-76], 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, 4e-290], 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, 1.2e-17], 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 < -4.9999999999999998e-76

   1. Initial program 95.8%

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

   3. Taylor expanded in 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -4.9999999999999998e-76 < t < 4.0000000000000003e-290

   1. Initial program 88.2%

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

   3. Taylor expanded in t around 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 98.0

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

   5. Applied rewrites 98.0%

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

   if 4.0000000000000003e-290 < t < 1.19999999999999993e-17

   1. Initial program 93.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if 1.19999999999999993e-17 < t

   1. Initial program 96.4%

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

   3. Taylor expanded in 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 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 7: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(\left(c + \frac{1}{\sqrt{a}} \cdot \frac{z}{t}\right) - b\right)\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -5e-76)
   (/
    x
    (+ x (* y (exp (* 2.0 (* a (- (+ c (* (/ 1.0 (sqrt a)) (/ z t))) b)))))))
   (if (<= t 2.6e-96)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) 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 <= -5e-76) {
		tmp = x / (x + (y * exp((2.0 * (a * ((c + ((1.0 / sqrt(a)) * (z / t))) - b))))));
	} else if (t <= 2.6e-96) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / 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 <= -5e-76)
		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 <= 2.6e-96)
		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(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, -5e-76], 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, 2.6e-96], 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[(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 3 regimes

2. if t < -4.9999999999999998e-76

   1. Initial program 95.8%

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

   3. Taylor expanded in 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -4.9999999999999998e-76 < t < 2.6000000000000002e-96

   1. Initial program 89.9%

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}} \]
   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 86.4

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

   5. Applied rewrites 86.4%

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

   if 2.6000000000000002e-96 < t

   1. Initial program 96.6%

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

   3. Taylor expanded in 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 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 8: 89.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(a \cdot \left(c - b\right)\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\mathsf{fma}\left(\sqrt{a}, z, 0.6666666666666666 \cdot \left(b - c\right)\right)}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{1}{\sqrt{t}} \cdot z - \left(0.8333333333333334 + a\right) \cdot \left(b - c\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -1e+26)
   (/ x (+ x (* y (exp (* 2.0 (* a (- c b)))))))
   (if (<= t 2.6e-96)
     (/
      x
      (+
       x
       (*
        y
        (exp (* 2.0 (/ (fma (sqrt a) z (* 0.6666666666666666 (- b c))) 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+26) {
		tmp = x / (x + (y * exp((2.0 * (a * (c - b))))));
	} else if (t <= 2.6e-96) {
		tmp = x / (x + (y * exp((2.0 * (fma(sqrt(a), z, (0.6666666666666666 * (b - c))) / 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+26)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(2.0 * Float64(a * Float64(c - b)))))));
	elseif (t <= 2.6e-96)
		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(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+26], 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, 2.6e-96], 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[(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 3 regimes

2. if t < -1.00000000000000005e26

   1. Initial program 94.7%

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

   3. Taylor expanded in 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 93.9

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

   8. Applied rewrites 93.9%

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

   if -1.00000000000000005e26 < t < 2.6000000000000002e-96

   1. Initial program 90.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\sqrt{a} \cdot z - \frac{-2}{3} \cdot \left(b - c\right)}{t}}}} \]
   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 86.9

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

   5. Applied rewrites 86.9%

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

   if 2.6000000000000002e-96 < t

   1. Initial program 96.6%

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

   3. Taylor expanded in 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 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 9: 51.8% 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 94.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 51.8%

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

Developer Target 1: 95.3% 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 2025091 
(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)))))))))))