Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 8

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

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, i)
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), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

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, i)
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), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
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, i)
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), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

assert x < y && y < z && z < t && t < a && a < b && b < c && c < i;
public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

[x, y, z, t, a, b, c, i] = sort([x, y, z, t, a, b, c, i])
def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

x, y, z, t, a, b, c, i = num2cell(sort([x, y, z, t, a, b, c, i])){:}
function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 95.0% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 4e-14)
   (+
    (+ (+ (+ (* (fma x (/ (log y) z) 1.0) z) t) a) (* (- b 0.5) (log c)))
    (* y i))
   (*
    (+
     (fma
      (* (- x) (/ (log y) y))
      -1.0
      (+ (+ (/ (fma (log c) (- b 0.5) z) y) (/ t y)) (/ a y)))
     i)
    y)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 4e-14) {
		tmp = ((((fma(x, (log(y) / z), 1.0) * z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	} else {
		tmp = (fma((-x * (log(y) / y)), -1.0, (((fma(log(c), (b - 0.5), z) / y) + (t / y)) + (a / y))) + i) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 4e-14)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(x, Float64(log(y) / z), 1.0) * z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-x) * Float64(log(y) / y)), -1.0, Float64(Float64(Float64(fma(log(c), Float64(b - 0.5), z) / y) + Float64(t / y)) + Float64(a / y))) + i) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 4e-14], N[(N[(N[(N[(N[(N[(x * N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-x) * N[(N[Log[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] / y), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e-14

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      7. lift-log.f64 93.5

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Applied rewrites 93.5%

      \[\leadsto \left(\left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   if 4e-14 < y

   1. Initial program 99.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot \frac{-\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(\left(\left(\left({x}^{-1} + \frac{\log y}{z}\right) \cdot x\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 9.5e-30)
   (+
    (+
     (+ (+ (* (* (+ (pow x -1.0) (/ (log y) z)) x) z) t) a)
     (* (- b 0.5) (log c)))
    (* y i))
   (*
    (+
     (fma
      (* (- x) (/ (log y) y))
      -1.0
      (+ (+ (/ (fma (log c) (- b 0.5) z) y) (/ t y)) (/ a y)))
     i)
    y)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 9.5e-30) {
		tmp = ((((((pow(x, -1.0) + (log(y) / z)) * x) * z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	} else {
		tmp = (fma((-x * (log(y) / y)), -1.0, (((fma(log(c), (b - 0.5), z) / y) + (t / y)) + (a / y))) + i) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 9.5e-30)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64((x ^ -1.0) + Float64(log(y) / z)) * x) * z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-x) * Float64(log(y) / y)), -1.0, Float64(Float64(Float64(fma(log(c), Float64(b - 0.5), z) / y) + Float64(t / y)) + Float64(a / y))) + i) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 9.5e-30], N[(N[(N[(N[(N[(N[(N[(N[Power[x, -1.0], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-x) * N[(N[Log[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] / y), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.49999999999999939e-30

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      7. lift-log.f64 93.2

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   5. Applied rewrites 93.2%

      \[\leadsto \left(\left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{\log y}{z}, 1\right) \cdot z} + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(\left(\left({x}^{0} + \frac{x \cdot \log y}{z}\right) \cdot z + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. inv-pow N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-log.f64 N/A

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

      19. lift-/.f64 93.2

          \[\leadsto \left(\left(\left(\left(\left({x}^{-1} + \frac{\log y}{z}\right) \cdot x\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   7. Applied rewrites 93.2%

      \[\leadsto \left(\left(\left(\left(\left({x}^{-1} + \frac{\log y}{z}\right) \cdot x\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]

   if 9.49999999999999939e-30 < y

   1. Initial program 99.2%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot \frac{-\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(\left(\left(\left({x}^{-1} + \frac{\log y}{z}\right) \cdot x\right) \cdot z + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-18}:\\ \;\;\;\;\left(\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z} - -1\right) \cdot z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(\log y, x, t + z\right)}{a} - -1\right) \cdot a + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -1.45e-18)
   (+
    (* (- (/ (+ (+ a t) (fma (log c) (- b 0.5) (* (log y) x))) z) -1.0) z)
    (* y i))
   (+
    (+ (* (- (/ (fma (log y) x (+ t z)) a) -1.0) a) (* (- b 0.5) (log c)))
    (* y i))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -1.45e-18) {
		tmp = (((((a + t) + fma(log(c), (b - 0.5), (log(y) * x))) / z) - -1.0) * z) + (y * i);
	} else {
		tmp = ((((fma(log(y), x, (t + z)) / a) - -1.0) * a) + ((b - 0.5) * log(c))) + (y * i);
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -1.45e-18)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a + t) + fma(log(c), Float64(b - 0.5), Float64(log(y) * x))) / z) - -1.0) * z) + Float64(y * i));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(log(y), x, Float64(t + z)) / a) - -1.0) * a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -1.45e-18], N[(N[(N[(N[(N[(N[(a + t), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - -1.0), $MachinePrecision] * z), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Log[y], $MachinePrecision] * x + N[(t + z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] - -1.0), $MachinePrecision] * a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e-18

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{a + \left(t + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)}{z} - 1\right)\right)\right) + y \cdot i \]

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\left(-\left(\left(-\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z}\right) - 1\right) \cdot z\right)} + y \cdot i \]

   if -1.45e-18 < z

   1. Initial program 99.3%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.0%

      \[\leadsto \left(\color{blue}{\left(\frac{\mathsf{fma}\left(\log y, x, t + z\right)}{a} + 1\right) \cdot a} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-18}:\\ \;\;\;\;\left(\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z} - -1\right) \cdot z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(\log y, x, t + z\right)}{a} - -1\right) \cdot a + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.9% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \mathsf{fma}\left(\log c, b - 0.5, t\_1\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{-56}:\\ \;\;\;\;\left(\frac{\left(a + t\right) + t\_2}{z} - -1\right) \cdot z + y \cdot i\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\mathsf{fma}\left(-0.5, \log c, t\_1\right) + z\right) + t\right) + a}{b}, -1, -\log c\right) \cdot \left(-b\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{t\_2}{t}\right) + \frac{z}{t}\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (fma (log c) (- b 0.5) t_1)))
   (if (<= z -3e-56)
     (+ (* (- (/ (+ (+ a t) t_2) z) -1.0) z) (* y i))
     (if (<= z 5.8e-146)
       (+
        (*
         (fma (/ (+ (+ (+ (fma -0.5 (log c) t_1) z) t) a) b) -1.0 (- (log c)))
         (- b))
        (* y i))
       (* (+ (+ 1.0 (/ a t)) (+ (fma i (/ y t) (/ t_2 t)) (/ z t))) t)))))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double t_2 = fma(log(c), (b - 0.5), t_1);
	double tmp;
	if (z <= -3e-56) {
		tmp = (((((a + t) + t_2) / z) - -1.0) * z) + (y * i);
	} else if (z <= 5.8e-146) {
		tmp = (fma(((((fma(-0.5, log(c), t_1) + z) + t) + a) / b), -1.0, -log(c)) * -b) + (y * i);
	} else {
		tmp = ((1.0 + (a / t)) + (fma(i, (y / t), (t_2 / t)) + (z / t))) * t;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	t_2 = fma(log(c), Float64(b - 0.5), t_1)
	tmp = 0.0
	if (z <= -3e-56)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a + t) + t_2) / z) - -1.0) * z) + Float64(y * i));
	elseif (z <= 5.8e-146)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(fma(-0.5, log(c), t_1) + z) + t) + a) / b), -1.0, Float64(-log(c))) * Float64(-b)) + Float64(y * i));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(a / t)) + Float64(fma(i, Float64(y / t), Float64(t_2 / t)) + Float64(z / t))) * t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[z, -3e-56], N[(N[(N[(N[(N[(N[(a + t), $MachinePrecision] + t$95$2), $MachinePrecision] / z), $MachinePrecision] - -1.0), $MachinePrecision] * z), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e-146], N[(N[(N[(N[(N[(N[(N[(N[(-0.5 * N[Log[c], $MachinePrecision] + t$95$1), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] / b), $MachinePrecision] * -1.0 + (-N[Log[c], $MachinePrecision])), $MachinePrecision] * (-b)), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(a / t), $MachinePrecision]), $MachinePrecision] + N[(N[(i * N[(y / t), $MachinePrecision] + N[(t$95$2 / t), $MachinePrecision]), $MachinePrecision] + N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.99999999999999989e-56

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{a + \left(t + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)}{z} - 1\right)\right)\right) + y \cdot i \]

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(-\left(\left(-\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z}\right) - 1\right) \cdot z\right)} + y \cdot i \]

   if -2.99999999999999989e-56 < z < 5.80000000000000022e-146

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot \left(-1 \cdot \log c + -1 \cdot \frac{a + \left(t + \left(z + \left(\frac{-1}{2} \cdot \log c + x \cdot \log y\right)\right)\right)}{b}\right)\right)\right) + y \cdot i \]

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(\frac{\left(\left(\mathsf{fma}\left(-0.5, \log c, \log y \cdot x\right) + z\right) + t\right) + a}{b}, -1, -\log c\right) \cdot b\right)} + y \cdot i \]

   if 5.80000000000000022e-146 < z

   1. Initial program 98.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-56}:\\ \;\;\;\;\left(\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z} - -1\right) \cdot z + y \cdot i\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(\mathsf{fma}\left(-0.5, \log c, \log y \cdot x\right) + z\right) + t\right) + a}{b}, -1, -\log c\right) \cdot \left(-b\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z} - -1\right) \cdot z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.8e-20)
   (+
    (* (- (/ (+ (+ a t) (fma (log c) (- b 0.5) (* (log y) x))) z) -1.0) z)
    (* y i))
   (*
    (+
     (fma
      (* (- x) (/ (log y) y))
      -1.0
      (+ (+ (/ (fma (log c) (- b 0.5) z) y) (/ t y)) (/ a y)))
     i)
    y)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.8e-20) {
		tmp = (((((a + t) + fma(log(c), (b - 0.5), (log(y) * x))) / z) - -1.0) * z) + (y * i);
	} else {
		tmp = (fma((-x * (log(y) / y)), -1.0, (((fma(log(c), (b - 0.5), z) / y) + (t / y)) + (a / y))) + i) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.8e-20)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a + t) + fma(log(c), Float64(b - 0.5), Float64(log(y) * x))) / z) - -1.0) * z) + Float64(y * i));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-x) * Float64(log(y) / y)), -1.0, Float64(Float64(Float64(fma(log(c), Float64(b - 0.5), z) / y) + Float64(t / y)) + Float64(a / y))) + i) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.8e-20], N[(N[(N[(N[(N[(N[(a + t), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - -1.0), $MachinePrecision] * z), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-x) * N[(N[Log[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] / y), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999987e-20

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{a + \left(t + \left(x \cdot \log y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)}{z} - 1\right)\right)\right) + y \cdot i \]

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(-\left(\left(-\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z}\right) - 1\right) \cdot z\right)} + y \cdot i \]

   if 1.79999999999999987e-20 < y

   1. Initial program 99.1%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot \frac{-\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{z} - -1\right) \cdot z + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 1.86e-65)
   (*
    (+
     (+ 1.0 (/ a t))
     (+ (fma i (/ y t) (/ (fma (log c) (- b 0.5) (* (log y) x)) t)) (/ z t)))
    t)
   (*
    (+
     (fma
      (* (- x) (/ (log y) y))
      -1.0
      (+ (+ (/ (fma (log c) (- b 0.5) z) y) (/ t y)) (/ a y)))
     i)
    y)))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 1.86e-65) {
		tmp = ((1.0 + (a / t)) + (fma(i, (y / t), (fma(log(c), (b - 0.5), (log(y) * x)) / t)) + (z / t))) * t;
	} else {
		tmp = (fma((-x * (log(y) / y)), -1.0, (((fma(log(c), (b - 0.5), z) / y) + (t / y)) + (a / y))) + i) * y;
	}
	return tmp;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 1.86e-65)
		tmp = Float64(Float64(Float64(1.0 + Float64(a / t)) + Float64(fma(i, Float64(y / t), Float64(fma(log(c), Float64(b - 0.5), Float64(log(y) * x)) / t)) + Float64(z / t))) * t);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-x) * Float64(log(y) / y)), -1.0, Float64(Float64(Float64(fma(log(c), Float64(b - 0.5), z) / y) + Float64(t / y)) + Float64(a / y))) + i) * y);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 1.86e-65], N[(N[(N[(1.0 + N[(a / t), $MachinePrecision]), $MachinePrecision] + N[(N[(i * N[(y / t), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(N[(N[((-x) * N[(N[Log[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] / y), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.86000000000000006e-65

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t} \]

   if 1.86000000000000006e-65 < y

   1. Initial program 99.2%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x \cdot \frac{-\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.86 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-x\right) \cdot \frac{\log y}{y}, -1, \left(\frac{\mathsf{fma}\left(\log c, b - 0.5, z\right)}{y} + \frac{t}{y}\right) + \frac{a}{y}\right) + i\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.6% accurate, N/A× speedup

Math

\[\begin{array}{l} [x, y, z, t, a, b, c, i] = \mathsf{sort}([x, y, z, t, a, b, c, i])\\ \\ \left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t \end{array} \]

FPCore

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
(FPCore (x y z t a b c i)
 :precision binary64
 (*
  (+
   (+ 1.0 (/ a t))
   (+ (fma i (/ y t) (/ (fma (log c) (- b 0.5) (* (log y) x)) t)) (/ z t)))
  t))

C

assert(x < y && y < z && z < t && t < a && a < b && b < c && c < i);
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((1.0 + (a / t)) + (fma(i, (y / t), (fma(log(c), (b - 0.5), (log(y) * x)) / t)) + (z / t))) * t;
}

Julia

x, y, z, t, a, b, c, i = sort([x, y, z, t, a, b, c, i])
function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(1.0 + Float64(a / t)) + Float64(fma(i, Float64(y / t), Float64(fma(log(c), Float64(b - 0.5), Float64(log(y) * x)) / t)) + Float64(z / t))) * t)
end

Wolfram

NOTE: x, y, z, t, a, b, c, and i should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(1.0 + N[(a / t), $MachinePrecision]), $MachinePrecision] + N[(N[(i * N[(y / t), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{a}{t}\right) + \left(\mathsf{fma}\left(i, \frac{y}{t}, \frac{\mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)}{t}\right) + \frac{z}{t}\right)\right) \cdot t} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025059 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))