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

Percentage Accurate: 93.7% → 96.4%

Time: 17.3s

Alternatives: 10

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: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 96.4% accurate, 0.5× speedup

Math

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

FPCore

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

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(((c + c) * (0.8333333333333334 + a)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z, t, a, b, c):
	t_1 = x / (x + (y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))))
	tmp = 0
	if t_1 <= 2.0:
		tmp = t_1
	else:
		tmp = x / (x + (y * math.exp(((c + c) * (0.8333333333333334 + a)))))
	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(Float64(c + c) * Float64(0.8333333333333334 + a))))));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x / N[(x + N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2.0], t$95$1, N[(x / N[(x + N[(y * N[Exp[N[(N[(c + c), $MachinePrecision] * N[(0.8333333333333334 + a), $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 98.7%

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

   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. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 59.3

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval 52.6

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

   7. Applied rewrites 52.6%

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

Alternative 2: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + y \cdot e^{\left(b + b\right) \cdot \left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right)}}\\ \mathbf{if}\;b \leq -1.36 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(c + c\right) \cdot \left(\left(0.8333333333333334 + a\right) - \frac{0.6666666666666666}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (/
          x
          (+
           x
           (*
            y
            (exp
             (*
              (+ b b)
              (- (- (/ 0.6666666666666666 t) a) 0.8333333333333334))))))))
   (if (<= b -1.36e+73)
     t_1
     (if (<= b 9.8e-82)
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            (+ c c)
            (- (+ 0.8333333333333334 a) (/ 0.6666666666666666 t)))))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x / (x + (y * exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))));
	double tmp;
	if (b <= -1.36e+73) {
		tmp = t_1;
	} else if (b <= 9.8e-82) {
		tmp = x / (x + (y * exp(((c + c) * ((0.8333333333333334 + a) - (0.6666666666666666 / t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (x + (y * exp(((b + b) * (((0.6666666666666666d0 / t) - a) - 0.8333333333333334d0)))))
    if (b <= (-1.36d+73)) then
        tmp = t_1
    else if (b <= 9.8d-82) then
        tmp = x / (x + (y * exp(((c + c) * ((0.8333333333333334d0 + a) - (0.6666666666666666d0 / t))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = x / (x + (y * Math.exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))));
	double tmp;
	if (b <= -1.36e+73) {
		tmp = t_1;
	} else if (b <= 9.8e-82) {
		tmp = x / (x + (y * Math.exp(((c + c) * ((0.8333333333333334 + a) - (0.6666666666666666 / t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = x / (x + (y * math.exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))))
	tmp = 0
	if b <= -1.36e+73:
		tmp = t_1
	elif b <= 9.8e-82:
		tmp = x / (x + (y * math.exp(((c + c) * ((0.8333333333333334 + a) - (0.6666666666666666 / t))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(b + b) * Float64(Float64(Float64(0.6666666666666666 / t) - a) - 0.8333333333333334))))))
	tmp = 0.0
	if (b <= -1.36e+73)
		tmp = t_1;
	elseif (b <= 9.8e-82)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(c + c) * Float64(Float64(0.8333333333333334 + a) - Float64(0.6666666666666666 / t)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = x / (x + (y * exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))));
	tmp = 0.0;
	if (b <= -1.36e+73)
		tmp = t_1;
	elseif (b <= 9.8e-82)
		tmp = x / (x + (y * exp(((c + c) * ((0.8333333333333334 + a) - (0.6666666666666666 / t))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(x / N[(x + N[(y * N[Exp[N[(N[(b + b), $MachinePrecision] * N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - a), $MachinePrecision] - 0.8333333333333334), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.36e+73], t$95$1, If[LessEqual[b, 9.8e-82], N[(x / N[(x + N[(y * N[Exp[N[(N[(c + c), $MachinePrecision] * N[(N[(0.8333333333333334 + a), $MachinePrecision] - N[(0.6666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3599999999999999e73 or 9.8000000000000006e-82 < b

   1. Initial program 91.8%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval 80.5

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

   4. Applied rewrites 80.5%

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

   if -1.3599999999999999e73 < b < 9.8000000000000006e-82

   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. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 75.8

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

   4. Applied rewrites 75.8%

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

Alternative 3: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)\\ \mathbf{if}\;t\_1 \leq -40000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+243}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\frac{\sqrt{a + t}}{t} \cdot \left(z + z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(b + b\right) \cdot \left(\left(\frac{0.6666666666666666}{t} - a\right) - 0.8333333333333334\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          2.0
          (-
           (/ (* z (sqrt (+ t a))) t)
           (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))
   (if (<= t_1 -40000000.0)
     1.0
     (if (<= t_1 2e+243)
       (/ x (+ x (* y (exp (* (/ (sqrt (+ a t)) t) (+ z z))))))
       (/
        x
        (+
         x
         (*
          y
          (exp
           (*
            (+ b b)
            (- (- (/ 0.6666666666666666 t) a) 0.8333333333333334))))))))))

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 <= -40000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 2e+243) {
		tmp = x / (x + (y * exp(((sqrt((a + t)) / t) * (z + z)))));
	} else {
		tmp = x / (x + (y * exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))
    if (t_1 <= (-40000000.0d0)) then
        tmp = 1.0d0
    else if (t_1 <= 2d+243) then
        tmp = x / (x + (y * exp(((sqrt((a + t)) / t) * (z + z)))))
    else
        tmp = x / (x + (y * exp(((b + b) * (((0.6666666666666666d0 / t) - a) - 0.8333333333333334d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))));
	double tmp;
	if (t_1 <= -40000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 2e+243) {
		tmp = x / (x + (y * Math.exp(((Math.sqrt((a + t)) / t) * (z + z)))));
	} else {
		tmp = x / (x + (y * Math.exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = 2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))
	tmp = 0
	if t_1 <= -40000000.0:
		tmp = 1.0
	elif t_1 <= 2e+243:
		tmp = x / (x + (y * math.exp(((math.sqrt((a + t)) / t) * (z + z)))))
	else:
		tmp = x / (x + (y * math.exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))))
	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 <= -40000000.0)
		tmp = 1.0;
	elseif (t_1 <= 2e+243)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(sqrt(Float64(a + t)) / t) * Float64(z + z))))));
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(b + b) * Float64(Float64(Float64(0.6666666666666666 / t) - a) - 0.8333333333333334))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = 2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))));
	tmp = 0.0;
	if (t_1 <= -40000000.0)
		tmp = 1.0;
	elseif (t_1 <= 2e+243)
		tmp = x / (x + (y * exp(((sqrt((a + t)) / t) * (z + z)))));
	else
		tmp = x / (x + (y * exp(((b + b) * (((0.6666666666666666 / t) - a) - 0.8333333333333334)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -40000000.0], 1.0, If[LessEqual[t$95$1, 2e+243], N[(x / N[(x + N[(y * N[Exp[N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + N[(y * N[Exp[N[(N[(b + b), $MachinePrecision] * N[(N[(N[(0.6666666666666666 / t), $MachinePrecision] - a), $MachinePrecision] - 0.8333333333333334), $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))))))) < -4e7

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.7%

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

   if -4e7 < (*.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))))))) < 2.0000000000000001e243

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. +-commutative N/A

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

      3. count-2-rev N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{z \cdot \sqrt{t + a}}{t} + \color{blue}{\frac{z \cdot \sqrt{t + a}}{t}}}} \]

      4. associate-/l* N/A

         \[\leadsto \frac{x}{x + y \cdot e^{z \cdot \frac{\sqrt{t + a}}{t} + \frac{\color{blue}{z \cdot \sqrt{t + a}}}{t}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{x}{x + y \cdot e^{z \cdot \frac{\sqrt{t + a}}{t} + z \cdot \color{blue}{\frac{\sqrt{t + a}}{t}}}} \]

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 58.1

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

   4. Applied rewrites 58.1%

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

   if 2.0000000000000001e243 < (*.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 83.1%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval 62.9

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

   4. Applied rewrites 62.9%

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

Alternative 4: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\frac{\sqrt{a + t}}{t} \cdot \left(z + z\right)}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(c + c\right) \cdot \left(0.8333333333333334 + a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          y
          (exp
           (*
            2.0
            (-
             (/ (* z (sqrt (+ t a))) t)
             (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))
   (if (<= t_1 -2e-111)
     (/ x (+ x (* y (exp (* (/ (sqrt (+ a t)) t) (+ z z))))))
     (if (<= t_1 0.0)
       1.0
       (/ x (+ x (* y (exp (* (+ c c) (+ 0.8333333333333334 a))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))));
	double tmp;
	if (t_1 <= -2e-111) {
		tmp = x / (x + (y * exp(((sqrt((a + t)) / t) * (z + z)))));
	} else if (t_1 <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * exp(((c + c) * (0.8333333333333334 + a)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * exp((2.0d0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0d0 / 6.0d0)) - (2.0d0 / (t * 3.0d0)))))))
    if (t_1 <= (-2d-111)) then
        tmp = x / (x + (y * exp(((sqrt((a + t)) / t) * (z + z)))))
    else if (t_1 <= 0.0d0) then
        tmp = 1.0d0
    else
        tmp = x / (x + (y * exp(((c + c) * (0.8333333333333334d0 + a)))))
    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 = y * Math.exp((2.0 * (((z * Math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))));
	double tmp;
	if (t_1 <= -2e-111) {
		tmp = x / (x + (y * Math.exp(((Math.sqrt((a + t)) / t) * (z + z)))));
	} else if (t_1 <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * Math.exp(((c + c) * (0.8333333333333334 + a)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = y * math.exp((2.0 * (((z * math.sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))))
	tmp = 0
	if t_1 <= -2e-111:
		tmp = x / (x + (y * math.exp(((math.sqrt((a + t)) / t) * (z + z)))))
	elif t_1 <= 0.0:
		tmp = 1.0
	else:
		tmp = x / (x + (y * math.exp(((c + c) * (0.8333333333333334 + a)))))
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))
	tmp = 0.0
	if (t_1 <= -2e-111)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(sqrt(Float64(a + t)) / t) * Float64(z + z))))));
	elseif (t_1 <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(c + c) * Float64(0.8333333333333334 + a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))));
	tmp = 0.0;
	if (t_1 <= -2e-111)
		tmp = x / (x + (y * exp(((sqrt((a + t)) / t) * (z + z)))));
	elseif (t_1 <= 0.0)
		tmp = 1.0;
	else
		tmp = x / (x + (y * exp(((c + c) * (0.8333333333333334 + a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-111], N[(x / N[(x + N[(y * N[Exp[N[(N[(N[Sqrt[N[(a + t), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], 1.0, N[(x / N[(x + N[(y * N[Exp[N[(N[(c + c), $MachinePrecision] * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))) < -2.00000000000000018e-111

   1. Initial program 98.4%

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

   2. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. +-commutative N/A

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

      3. count-2-rev N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{z \cdot \sqrt{t + a}}{t} + \color{blue}{\frac{z \cdot \sqrt{t + a}}{t}}}} \]

      4. associate-/l* N/A

         \[\leadsto \frac{x}{x + y \cdot e^{z \cdot \frac{\sqrt{t + a}}{t} + \frac{\color{blue}{z \cdot \sqrt{t + a}}}{t}}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{x}{x + y \cdot e^{z \cdot \frac{\sqrt{t + a}}{t} + z \cdot \color{blue}{\frac{\sqrt{t + a}}{t}}}} \]

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 54.8

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

   4. Applied rewrites 54.8%

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

   if -2.00000000000000018e-111 < (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))) < 0.0

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.5%

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

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

   1. Initial program 82.5%

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

   2. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval 54.6

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

   7. Applied rewrites 54.6%

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

Alternative 5: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\frac{c}{t} \cdot -1.3333333333333333}, y, x\right)}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(c + c\right) \cdot \left(0.8333333333333334 + a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          y
          (exp
           (*
            2.0
            (-
             (/ (* z (sqrt (+ t a))) t)
             (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))
   (if (<= t_1 -2e-111)
     (/ x (fma (exp (* (/ c t) -1.3333333333333333)) y x))
     (if (<= t_1 0.0)
       1.0
       (/ x (+ x (* y (exp (* (+ c c) (+ 0.8333333333333334 a))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = y * exp((2.0 * (((z * sqrt((t + a))) / t) - ((b - c) * ((a + (5.0 / 6.0)) - (2.0 / (t * 3.0)))))));
	double tmp;
	if (t_1 <= -2e-111) {
		tmp = x / fma(exp(((c / t) * -1.3333333333333333)), y, x);
	} else if (t_1 <= 0.0) {
		tmp = 1.0;
	} else {
		tmp = x / (x + (y * exp(((c + c) * (0.8333333333333334 + a)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(y * exp(Float64(2.0 * Float64(Float64(Float64(z * sqrt(Float64(t + a))) / t) - Float64(Float64(b - c) * Float64(Float64(a + Float64(5.0 / 6.0)) - Float64(2.0 / Float64(t * 3.0))))))))
	tmp = 0.0
	if (t_1 <= -2e-111)
		tmp = Float64(x / fma(exp(Float64(Float64(c / t) * -1.3333333333333333)), y, x));
	elseif (t_1 <= 0.0)
		tmp = 1.0;
	else
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(c + c) * Float64(0.8333333333333334 + a))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * N[Exp[N[(2.0 * N[(N[(N[(z * N[Sqrt[N[(t + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(N[(b - c), $MachinePrecision] * N[(N[(a + N[(5.0 / 6.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(t * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-111], N[(x / N[(N[Exp[N[(N[(c / t), $MachinePrecision] * -1.3333333333333333), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], 1.0, N[(x / N[(x + N[(y * N[Exp[N[(N[(c + c), $MachinePrecision] * N[(0.8333333333333334 + a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))) < -2.00000000000000018e-111

   1. Initial program 98.4%

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

   2. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{-4}{3} \cdot \color{blue}{\frac{c}{t}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \frac{-4}{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \frac{-4}{3}}} \]

      3. lower-/.f64 44.6

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}} \]

   7. Applied rewrites 44.6%

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \color{blue}{-1.3333333333333333}}} \]
   8. Step-by-step derivation

      1. metadata-eval 44.6

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}} \]

   9. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(e^{\frac{c}{t} \cdot -1.3333333333333333}, y, x\right)}} \]

   if -2.00000000000000018e-111 < (*.f64 y (exp.f64 (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64))))))))) < 0.0

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.5%

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

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

   1. Initial program 82.5%

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

   2. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval 54.6

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

   7. Applied rewrites 54.6%

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

Alternative 6: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)\\ \mathbf{if}\;t\_1 \leq -40000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{\left(c + c\right) \cdot 0.8333333333333334}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\frac{c}{t} \cdot -1.3333333333333333}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          2.0
          (-
           (/ (* z (sqrt (+ t a))) t)
           (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))
   (if (<= t_1 -40000000.0)
     1.0
     (if (<= t_1 2e+287)
       (/ x (+ x (* y (exp (* (+ c c) 0.8333333333333334)))))
       (/ x (fma (exp (* (/ c t) -1.3333333333333333)) y x))))))

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 <= -40000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 2e+287) {
		tmp = x / (x + (y * exp(((c + c) * 0.8333333333333334))));
	} else {
		tmp = x / fma(exp(((c / t) * -1.3333333333333333)), y, x);
	}
	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 <= -40000000.0)
		tmp = 1.0;
	elseif (t_1 <= 2e+287)
		tmp = Float64(x / Float64(x + Float64(y * exp(Float64(Float64(c + c) * 0.8333333333333334)))));
	else
		tmp = Float64(x / fma(exp(Float64(Float64(c / t) * -1.3333333333333333)), y, x));
	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, -40000000.0], 1.0, If[LessEqual[t$95$1, 2e+287], N[(x / N[(x + N[(y * N[Exp[N[(N[(c + c), $MachinePrecision] * 0.8333333333333334), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Exp[N[(N[(c / t), $MachinePrecision] * -1.3333333333333333), $MachinePrecision]], $MachinePrecision] * y + x), $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))))))) < -4e7

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.7%

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

   if -4e7 < (*.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))))))) < 2.0000000000000002e287

   1. Initial program 99.9%

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

   2. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 66.0

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. metadata-eval 58.4

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

   7. Applied rewrites 58.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.1%

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

   if 2.0000000000000002e287 < (*.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 80.1%

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

   2. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 64.2

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

   4. Applied rewrites 64.2%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{-4}{3} \cdot \color{blue}{\frac{c}{t}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \frac{-4}{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \frac{-4}{3}}} \]

      3. lower-/.f64 49.6

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}} \]

   7. Applied rewrites 49.6%

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \color{blue}{-1.3333333333333333}}} \]
   8. Step-by-step derivation

      1. metadata-eval 49.6

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}} \]

   9. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(e^{\frac{c}{t} \cdot -1.3333333333333333}, y, x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)\\ \mathbf{if}\;t\_1 \leq -40000000:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{-1.6666666666666667 \cdot b}, y, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\frac{b}{t} \cdot 1.3333333333333333}, y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1
         (*
          2.0
          (-
           (/ (* z (sqrt (+ t a))) t)
           (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))
   (if (<= t_1 -40000000.0)
     1.0
     (if (<= t_1 2e+298)
       (/ x (fma (exp (* -1.6666666666666667 b)) y x))
       (/ x (fma (exp (* (/ b t) 1.3333333333333333)) y x))))))

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 <= -40000000.0) {
		tmp = 1.0;
	} else if (t_1 <= 2e+298) {
		tmp = x / fma(exp((-1.6666666666666667 * b)), y, x);
	} else {
		tmp = x / fma(exp(((b / t) * 1.3333333333333333)), y, x);
	}
	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 <= -40000000.0)
		tmp = 1.0;
	elseif (t_1 <= 2e+298)
		tmp = Float64(x / fma(exp(Float64(-1.6666666666666667 * b)), y, x));
	else
		tmp = Float64(x / fma(exp(Float64(Float64(b / t) * 1.3333333333333333)), y, x));
	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, -40000000.0], 1.0, If[LessEqual[t$95$1, 2e+298], N[(x / N[(N[Exp[N[(-1.6666666666666667 * b), $MachinePrecision]], $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Exp[N[(N[(b / t), $MachinePrecision] * 1.3333333333333333), $MachinePrecision]], $MachinePrecision] * y + x), $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))))))) < -4e7

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.7%

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

   if -4e7 < (*.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))))))) < 1.9999999999999999e298

   1. Initial program 99.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval 67.0

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval 58.5

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

   7. Applied rewrites 58.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{-5}{3} \cdot b}} \]
   9. Step-by-step derivation

      1. lower-*.f64 50.9

         \[\leadsto \frac{x}{x + y \cdot e^{-1.6666666666666667 \cdot b}} \]

   10. Applied rewrites 50.9%

       \[\leadsto \frac{x}{x + y \cdot e^{-1.6666666666666667 \cdot b}} \]

   12. Applied rewrites 50.9%

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

   if 1.9999999999999999e298 < (*.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 79.2%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval 62.9

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{4}{3} \cdot \color{blue}{\frac{b}{t}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{b}{t} \cdot \frac{4}{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{b}{t} \cdot \frac{4}{3}}} \]

      3. lower-/.f64 48.9

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{b}{t} \cdot 1.3333333333333333}} \]

   7. Applied rewrites 48.9%

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{b}{t} \cdot \color{blue}{1.3333333333333333}}} \]
   8. Step-by-step derivation

      1. metadata-eval 48.9

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{b}{t} \cdot 1.3333333333333333}} \]

   9. Applied rewrites 48.9%

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

Alternative 8: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) \leq -40000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{\frac{c}{t} \cdot -1.3333333333333333}, y, x\right)}\\ \end{array} \end{array} \]

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))))))
      -40000000.0)
   1.0
   (/ x (fma (exp (* (/ c t) -1.3333333333333333)) y x))))

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)))))) <= -40000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp(((c / t) * -1.3333333333333333)), y, x);
	}
	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)))))) <= -40000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(Float64(c / t) * -1.3333333333333333)), y, x));
	end
	return 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], -40000000.0], 1.0, N[(x / N[(N[Exp[N[(N[(c / t), $MachinePrecision] * -1.3333333333333333), $MachinePrecision]], $MachinePrecision] * y + x), $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))))))) < -4e7

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.7%

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

   if -4e7 < (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))

   1. Initial program 89.9%

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

   2. Taylor expanded in c around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 65.1

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

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{-4}{3} \cdot \color{blue}{\frac{c}{t}}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \frac{-4}{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \frac{-4}{3}}} \]

      3. lower-/.f64 47.5

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}} \]

   7. Applied rewrites 47.5%

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot \color{blue}{-1.3333333333333333}}} \]
   8. Step-by-step derivation

      1. metadata-eval 47.5

         \[\leadsto \frac{x}{x + y \cdot e^{\frac{c}{t} \cdot -1.3333333333333333}} \]

   9. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(e^{\frac{c}{t} \cdot -1.3333333333333333}, y, x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) \leq -40000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(e^{-1.6666666666666667 \cdot b}, y, x\right)}\\ \end{array} \end{array} \]

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))))))
      -40000000.0)
   1.0
   (/ x (fma (exp (* -1.6666666666666667 b)) y x))))

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)))))) <= -40000000.0) {
		tmp = 1.0;
	} else {
		tmp = x / fma(exp((-1.6666666666666667 * b)), y, x);
	}
	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)))))) <= -40000000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / fma(exp(Float64(-1.6666666666666667 * b)), y, x));
	end
	return 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], -40000000.0], 1.0, N[(x / N[(N[Exp[N[(-1.6666666666666667 * b), $MachinePrecision]], $MachinePrecision] * y + x), $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))))))) < -4e7

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 98.7%

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

   if -4e7 < (*.f64 #s(literal 2 binary64) (-.f64 (/.f64 (*.f64 z (sqrt.f64 (+.f64 t a))) t) (*.f64 (-.f64 b c) (-.f64 (+.f64 a (/.f64 #s(literal 5 binary64) #s(literal 6 binary64))) (/.f64 #s(literal 2 binary64) (*.f64 t #s(literal 3 binary64)))))))

   1. Initial program 89.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in t around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{x}{x + y \cdot e^{\frac{-5}{3} \cdot b}} \]
   9. Step-by-step derivation

      1. lower-*.f64 47.6

         \[\leadsto \frac{x}{x + y \cdot e^{-1.6666666666666667 \cdot b}} \]

   10. Applied rewrites 47.6%

       \[\leadsto \frac{x}{x + y \cdot e^{-1.6666666666666667 \cdot b}} \]

   12. Applied rewrites 47.6%

       \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(e^{-1.6666666666666667 \cdot b}, y, x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 50.8% accurate, 57.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :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)))))))))))