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

Percentage Accurate: 99.8% → 99.8%

Time: 1.3min

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in i around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 34.2%

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

5. Taylor expanded in b around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-log.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 86.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) + \log y\right) \cdot x\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(y + \frac{a}{i}\right) + \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{i}, \frac{\log y \cdot x}{i}\right) + \frac{z}{i}\right) + \frac{t}{i}\right)\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (*
          (+
           (+
            (+ (/ (+ t z) x) (fma (log c) (/ (- b 0.5) x) (/ (* i y) x)))
            (/ a x))
           (log y))
          x)))
   (if (<= x -6.4e-86)
     t_1
     (if (<= x 1.15e-39)
       (*
        (+
         (+ y (/ a i))
         (+
          (+ (fma (log c) (/ (- b 0.5) i) (/ (* (log y) x) i)) (/ z i))
          (/ t i)))
        i)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((t + z) / x) + fma(log(c), ((b - 0.5) / x), ((i * y) / x))) + (a / x)) + log(y)) * x;
	double tmp;
	if (x <= -6.4e-86) {
		tmp = t_1;
	} else if (x <= 1.15e-39) {
		tmp = ((y + (a / i)) + ((fma(log(c), ((b - 0.5) / i), ((log(y) * x) / i)) + (z / i)) + (t / i))) * i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(t + z) / x) + fma(log(c), Float64(Float64(b - 0.5) / x), Float64(Float64(i * y) / x))) + Float64(a / x)) + log(y)) * x)
	tmp = 0.0
	if (x <= -6.4e-86)
		tmp = t_1;
	elseif (x <= 1.15e-39)
		tmp = Float64(Float64(Float64(y + Float64(a / i)) + Float64(Float64(fma(log(c), Float64(Float64(b - 0.5) / i), Float64(Float64(log(y) * x) / i)) + Float64(z / i)) + Float64(t / i))) * i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(t + z), $MachinePrecision] / x), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(N[(b - 0.5), $MachinePrecision] / x), $MachinePrecision] + N[(N[(i * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / x), $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.4e-86], t$95$1, If[LessEqual[x, 1.15e-39], N[(N[(N[(y + N[(a / i), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(N[(b - 0.5), $MachinePrecision] / i), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(z / i), $MachinePrecision]), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.40000000000000011e-86 or 1.15000000000000004e-39 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 96.8%

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

   if -6.40000000000000011e-86 < x < 1.15000000000000004e-39

   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. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 70.8%

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

Alternative 3: 84.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\log c}{x}\\ t_2 := \left(\left(y + \frac{a}{i}\right) + \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{i}, \frac{\log y \cdot x}{i}\right) + \frac{z}{i}\right) + \frac{t}{i}\right)\right) \cdot i\\ \mathbf{if}\;i \leq -4.6 \cdot 10^{-32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq 1.08 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\left(\frac{t + z}{x} + -1 \cdot \left(b \cdot \mathsf{fma}\left(-1, t\_1, -1 \cdot \frac{\mathsf{fma}\left(-0.5, t\_1, \frac{i \cdot y}{x}\right)}{b}\right)\right)\right) + \frac{a}{x}\right) + \log y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ (log c) x))
        (t_2
         (*
          (+
           (+ y (/ a i))
           (+
            (+ (fma (log c) (/ (- b 0.5) i) (/ (* (log y) x) i)) (/ z i))
            (/ t i)))
          i)))
   (if (<= i -4.6e-32)
     t_2
     (if (<= i 1.08e-32)
       (*
        (+
         (+
          (+
           (/ (+ t z) x)
           (*
            -1.0
            (* b (fma -1.0 t_1 (* -1.0 (/ (fma -0.5 t_1 (/ (* i y) x)) b))))))
          (/ a x))
         (log y))
        x)
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) / x;
	double t_2 = ((y + (a / i)) + ((fma(log(c), ((b - 0.5) / i), ((log(y) * x) / i)) + (z / i)) + (t / i))) * i;
	double tmp;
	if (i <= -4.6e-32) {
		tmp = t_2;
	} else if (i <= 1.08e-32) {
		tmp = (((((t + z) / x) + (-1.0 * (b * fma(-1.0, t_1, (-1.0 * (fma(-0.5, t_1, ((i * y) / x)) / b)))))) + (a / x)) + log(y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) / x)
	t_2 = Float64(Float64(Float64(y + Float64(a / i)) + Float64(Float64(fma(log(c), Float64(Float64(b - 0.5) / i), Float64(Float64(log(y) * x) / i)) + Float64(z / i)) + Float64(t / i))) * i)
	tmp = 0.0
	if (i <= -4.6e-32)
		tmp = t_2;
	elseif (i <= 1.08e-32)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(t + z) / x) + Float64(-1.0 * Float64(b * fma(-1.0, t_1, Float64(-1.0 * Float64(fma(-0.5, t_1, Float64(Float64(i * y) / x)) / b)))))) + Float64(a / x)) + log(y)) * x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y + N[(a / i), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(N[(b - 0.5), $MachinePrecision] / i), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(z / i), $MachinePrecision]), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -4.6e-32], t$95$2, If[LessEqual[i, 1.08e-32], N[(N[(N[(N[(N[(N[(t + z), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 * N[(b * N[(-1.0 * t$95$1 + N[(-1.0 * N[(N[(-0.5 * t$95$1 + N[(N[(i * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / x), $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if i < -4.6000000000000001e-32 or 1.08e-32 < i

   1. Initial program 99.7%

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

   2. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if -4.6000000000000001e-32 < i < 1.08e-32

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 66.1%

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

Alternative 4: 73.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ t_2 := \frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{t\_1}{x}\right)\right)\\ t_3 := \left|\log y\right|\\ t_4 := \left(\left(y + \frac{a}{i}\right) + \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{i}, \frac{\log y \cdot x}{i}\right) + \frac{z}{i}\right) + \frac{t}{i}\right)\right) \cdot i\\ \mathbf{if}\;i \leq -2.45 \cdot 10^{-119}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;i \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;\left(\left(z \cdot \left({x}^{-1} + \left(\frac{t}{x \cdot z} + \frac{\mathsf{fma}\left(i \cdot y, x \cdot z, \left(x \cdot z\right) \cdot t\_1\right)}{\left(x \cdot z\right) \cdot \left(x \cdot z\right)}\right)\right) + \frac{a}{x}\right) + \log y\right) \cdot x\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\log y}^{2}, \log y, {t\_2}^{3}\right)}{\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2\right) - \log y \cdot t\_2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) (- b 0.5)))
        (t_2 (+ (/ a x) (+ (/ t x) (+ (/ z x) (/ t_1 x)))))
        (t_3 (fabs (log y)))
        (t_4
         (*
          (+
           (+ y (/ a i))
           (+
            (+ (fma (log c) (/ (- b 0.5) i) (/ (* (log y) x) i)) (/ z i))
            (/ t i)))
          i)))
   (if (<= i -2.45e-119)
     t_4
     (if (<= i -4.5e-205)
       (*
        (+
         (+
          (*
           z
           (+
            (pow x -1.0)
            (+
             (/ t (* x z))
             (/ (fma (* i y) (* x z) (* (* x z) t_1)) (* (* x z) (* x z))))))
          (/ a x))
         (log y))
        x)
       (if (<= i 4.5e-74)
         (*
          (/
           (fma (pow (log y) 2.0) (log y) (pow t_2 3.0))
           (- (fma t_3 t_3 (* t_2 t_2)) (* (log y) t_2)))
          x)
         t_4)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * (b - 0.5);
	double t_2 = (a / x) + ((t / x) + ((z / x) + (t_1 / x)));
	double t_3 = fabs(log(y));
	double t_4 = ((y + (a / i)) + ((fma(log(c), ((b - 0.5) / i), ((log(y) * x) / i)) + (z / i)) + (t / i))) * i;
	double tmp;
	if (i <= -2.45e-119) {
		tmp = t_4;
	} else if (i <= -4.5e-205) {
		tmp = (((z * (pow(x, -1.0) + ((t / (x * z)) + (fma((i * y), (x * z), ((x * z) * t_1)) / ((x * z) * (x * z)))))) + (a / x)) + log(y)) * x;
	} else if (i <= 4.5e-74) {
		tmp = (fma(pow(log(y), 2.0), log(y), pow(t_2, 3.0)) / (fma(t_3, t_3, (t_2 * t_2)) - (log(y) * t_2))) * x;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * Float64(b - 0.5))
	t_2 = Float64(Float64(a / x) + Float64(Float64(t / x) + Float64(Float64(z / x) + Float64(t_1 / x))))
	t_3 = abs(log(y))
	t_4 = Float64(Float64(Float64(y + Float64(a / i)) + Float64(Float64(fma(log(c), Float64(Float64(b - 0.5) / i), Float64(Float64(log(y) * x) / i)) + Float64(z / i)) + Float64(t / i))) * i)
	tmp = 0.0
	if (i <= -2.45e-119)
		tmp = t_4;
	elseif (i <= -4.5e-205)
		tmp = Float64(Float64(Float64(Float64(z * Float64((x ^ -1.0) + Float64(Float64(t / Float64(x * z)) + Float64(fma(Float64(i * y), Float64(x * z), Float64(Float64(x * z) * t_1)) / Float64(Float64(x * z) * Float64(x * z)))))) + Float64(a / x)) + log(y)) * x);
	elseif (i <= 4.5e-74)
		tmp = Float64(Float64(fma((log(y) ^ 2.0), log(y), (t_2 ^ 3.0)) / Float64(fma(t_3, t_3, Float64(t_2 * t_2)) - Float64(log(y) * t_2))) * x);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / x), $MachinePrecision] + N[(N[(t / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[Log[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y + N[(a / i), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(N[(b - 0.5), $MachinePrecision] / i), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(z / i), $MachinePrecision]), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[i, -2.45e-119], t$95$4, If[LessEqual[i, -4.5e-205], N[(N[(N[(N[(z * N[(N[Power[x, -1.0], $MachinePrecision] + N[(N[(t / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(i * y), $MachinePrecision] * N[(x * z), $MachinePrecision] + N[(N[(x * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(x * z), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / x), $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[i, 4.5e-74], N[(N[(N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * t$95$3 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.45e-119 or 4.4999999999999999e-74 < i

   1. Initial program 99.7%

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

   2. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.7%

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

   if -2.45e-119 < i < -4.49999999999999956e-205

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

   10. Applied rewrites 36.4%

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

   if -4.49999999999999956e-205 < i < 4.4999999999999999e-74

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 35.9%

      \[\leadsto \frac{{\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right)}^{3} + {\log y}^{3}}{\mathsf{fma}\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \log y \cdot \log y - \left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) \cdot \log y\right)} \cdot x \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{{\log y}^{3} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{3}}{\left({\log y}^{2} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{2}\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.2%

      \[\leadsto \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\left|\log y\right|, \left|\log y\right|, \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right) \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\log y\right|\\ t_2 := \left(\left(y + \frac{a}{i}\right) + \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{i}, \frac{\log y \cdot x}{i}\right) + \frac{z}{i}\right) + \frac{t}{i}\right)\right) \cdot i\\ t_3 := \frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\\ \mathbf{if}\;i \leq -1.7 \cdot 10^{-167}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\log y}^{2}, \log y, {t\_3}^{3}\right)}{\mathsf{fma}\left(t\_1, t\_1, t\_3 \cdot t\_3\right) - \log y \cdot t\_3} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fabs (log y)))
        (t_2
         (*
          (+
           (+ y (/ a i))
           (+
            (+ (fma (log c) (/ (- b 0.5) i) (/ (* (log y) x) i)) (/ z i))
            (/ t i)))
          i))
        (t_3 (+ (/ a x) (+ (/ t x) (+ (/ z x) (/ (* (log c) (- b 0.5)) x))))))
   (if (<= i -1.7e-167)
     t_2
     (if (<= i 4.5e-74)
       (*
        (/
         (fma (pow (log y) 2.0) (log y) (pow t_3 3.0))
         (- (fma t_1 t_1 (* t_3 t_3)) (* (log y) t_3)))
        x)
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fabs(log(y));
	double t_2 = ((y + (a / i)) + ((fma(log(c), ((b - 0.5) / i), ((log(y) * x) / i)) + (z / i)) + (t / i))) * i;
	double t_3 = (a / x) + ((t / x) + ((z / x) + ((log(c) * (b - 0.5)) / x)));
	double tmp;
	if (i <= -1.7e-167) {
		tmp = t_2;
	} else if (i <= 4.5e-74) {
		tmp = (fma(pow(log(y), 2.0), log(y), pow(t_3, 3.0)) / (fma(t_1, t_1, (t_3 * t_3)) - (log(y) * t_3))) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = abs(log(y))
	t_2 = Float64(Float64(Float64(y + Float64(a / i)) + Float64(Float64(fma(log(c), Float64(Float64(b - 0.5) / i), Float64(Float64(log(y) * x) / i)) + Float64(z / i)) + Float64(t / i))) * i)
	t_3 = Float64(Float64(a / x) + Float64(Float64(t / x) + Float64(Float64(z / x) + Float64(Float64(log(c) * Float64(b - 0.5)) / x))))
	tmp = 0.0
	if (i <= -1.7e-167)
		tmp = t_2;
	elseif (i <= 4.5e-74)
		tmp = Float64(Float64(fma((log(y) ^ 2.0), log(y), (t_3 ^ 3.0)) / Float64(fma(t_1, t_1, Float64(t_3 * t_3)) - Float64(log(y) * t_3))) * x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Abs[N[Log[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y + N[(a / i), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[c], $MachinePrecision] * N[(N[(b - 0.5), $MachinePrecision] / i), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(z / i), $MachinePrecision]), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a / x), $MachinePrecision] + N[(N[(t / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.7e-167], t$95$2, If[LessEqual[i, 4.5e-74], N[(N[(N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.6999999999999999e-167 or 4.4999999999999999e-74 < i

   1. Initial program 99.7%

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

   2. Taylor expanded in i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 89.2%

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

   if -1.6999999999999999e-167 < i < 4.4999999999999999e-74

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   6. Applied rewrites 36.1%

      \[\leadsto \frac{{\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right)}^{3} + {\log y}^{3}}{\mathsf{fma}\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \log y \cdot \log y - \left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) \cdot \log y\right)} \cdot x \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{{\log y}^{3} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{3}}{\left({\log y}^{2} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{2}\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\left|\log y\right|, \left|\log y\right|, \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right) \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\log y\right|\\ t_2 := \left(-1 \cdot i\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-a, i, \left(-1 \cdot i\right) \cdot \left(\left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\right)\right)}{\left(-1 \cdot i\right) \cdot i}, -1, -1 \cdot y\right)\\ t_3 := \log c \cdot \left(b - 0.5\right)\\ t_4 := \left(-1 \cdot i\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{a}{i}, -1 \cdot \frac{t + \left(z + t\_3\right)}{i}\right)}{x}, \frac{\log y}{i}\right), -1, -1 \cdot y\right)\\ t_5 := \frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{t\_3}{x}\right)\right)\\ \mathbf{if}\;i \leq -500000:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;i \leq -1.05 \cdot 10^{-149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\log y}^{2}, \log y, {t\_5}^{3}\right)}{\mathsf{fma}\left(t\_1, t\_1, t\_5 \cdot t\_5\right) - \log y \cdot t\_5} \cdot x\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fabs (log y)))
        (t_2
         (*
          (* -1.0 i)
          (fma
           (/
            (fma
             (- a)
             i
             (* (* -1.0 i) (+ (+ t z) (fma (log c) (- b 0.5) (* (log y) x)))))
            (* (* -1.0 i) i))
           -1.0
           (* -1.0 y))))
        (t_3 (* (log c) (- b 0.5)))
        (t_4
         (*
          (* -1.0 i)
          (fma
           (*
            x
            (fma
             -1.0
             (/ (fma -1.0 (/ a i) (* -1.0 (/ (+ t (+ z t_3)) i))) x)
             (/ (log y) i)))
           -1.0
           (* -1.0 y))))
        (t_5 (+ (/ a x) (+ (/ t x) (+ (/ z x) (/ t_3 x))))))
   (if (<= i -500000.0)
     t_4
     (if (<= i -1.05e-149)
       t_2
       (if (<= i 4.5e-74)
         (*
          (/
           (fma (pow (log y) 2.0) (log y) (pow t_5 3.0))
           (- (fma t_1 t_1 (* t_5 t_5)) (* (log y) t_5)))
          x)
         (if (<= i 8.5e+47) t_2 t_4))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fabs(log(y));
	double t_2 = (-1.0 * i) * fma((fma(-a, i, ((-1.0 * i) * ((t + z) + fma(log(c), (b - 0.5), (log(y) * x))))) / ((-1.0 * i) * i)), -1.0, (-1.0 * y));
	double t_3 = log(c) * (b - 0.5);
	double t_4 = (-1.0 * i) * fma((x * fma(-1.0, (fma(-1.0, (a / i), (-1.0 * ((t + (z + t_3)) / i))) / x), (log(y) / i))), -1.0, (-1.0 * y));
	double t_5 = (a / x) + ((t / x) + ((z / x) + (t_3 / x)));
	double tmp;
	if (i <= -500000.0) {
		tmp = t_4;
	} else if (i <= -1.05e-149) {
		tmp = t_2;
	} else if (i <= 4.5e-74) {
		tmp = (fma(pow(log(y), 2.0), log(y), pow(t_5, 3.0)) / (fma(t_1, t_1, (t_5 * t_5)) - (log(y) * t_5))) * x;
	} else if (i <= 8.5e+47) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = abs(log(y))
	t_2 = Float64(Float64(-1.0 * i) * fma(Float64(fma(Float64(-a), i, Float64(Float64(-1.0 * i) * Float64(Float64(t + z) + fma(log(c), Float64(b - 0.5), Float64(log(y) * x))))) / Float64(Float64(-1.0 * i) * i)), -1.0, Float64(-1.0 * y)))
	t_3 = Float64(log(c) * Float64(b - 0.5))
	t_4 = Float64(Float64(-1.0 * i) * fma(Float64(x * fma(-1.0, Float64(fma(-1.0, Float64(a / i), Float64(-1.0 * Float64(Float64(t + Float64(z + t_3)) / i))) / x), Float64(log(y) / i))), -1.0, Float64(-1.0 * y)))
	t_5 = Float64(Float64(a / x) + Float64(Float64(t / x) + Float64(Float64(z / x) + Float64(t_3 / x))))
	tmp = 0.0
	if (i <= -500000.0)
		tmp = t_4;
	elseif (i <= -1.05e-149)
		tmp = t_2;
	elseif (i <= 4.5e-74)
		tmp = Float64(Float64(fma((log(y) ^ 2.0), log(y), (t_5 ^ 3.0)) / Float64(fma(t_1, t_1, Float64(t_5 * t_5)) - Float64(log(y) * t_5))) * x);
	elseif (i <= 8.5e+47)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Abs[N[Log[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 * i), $MachinePrecision] * N[(N[(N[((-a) * i + N[(N[(-1.0 * i), $MachinePrecision] * N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 * i), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 * i), $MachinePrecision] * N[(N[(x * N[(-1.0 * N[(N[(-1.0 * N[(a / i), $MachinePrecision] + N[(-1.0 * N[(N[(t + N[(z + t$95$3), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(a / x), $MachinePrecision] + N[(N[(t / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -500000.0], t$95$4, If[LessEqual[i, -1.05e-149], t$95$2, If[LessEqual[i, 4.5e-74], N[(N[(N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1 + N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[i, 8.5e+47], t$95$2, t$95$4]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -5e5 or 8.5000000000000008e47 < i

   1. Initial program 99.7%

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

   2. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 91.2%

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

   if -5e5 < i < -1.05000000000000005e-149 or 4.4999999999999999e-74 < i < 8.5000000000000008e47

   1. Initial program 99.8%

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

   2. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 69.9%

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

   if -1.05000000000000005e-149 < i < 4.4999999999999999e-74

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.5%

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

   6. Applied rewrites 36.2%

      \[\leadsto \frac{{\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right)}^{3} + {\log y}^{3}}{\mathsf{fma}\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \log y \cdot \log y - \left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) \cdot \log y\right)} \cdot x \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{{\log y}^{3} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{3}}{\left({\log y}^{2} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{2}\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\left|\log y\right|, \left|\log y\right|, \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right) \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)} \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\log y\right|\\ t_2 := \log c \cdot \left(b - 0.5\right)\\ t_3 := \frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{t\_2}{x}\right)\right)\\ t_4 := \left(-1 \cdot i\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \frac{a}{i}, -1 \cdot \frac{t + \left(z + t\_2\right)}{i}\right)}{x}, \frac{\log y}{i}\right), -1, -1 \cdot y\right)\\ \mathbf{if}\;i \leq -4.8 \cdot 10^{-51}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;i \leq 5.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\log y}^{2}, \log y, {t\_3}^{3}\right)}{\mathsf{fma}\left(t\_1, t\_1, t\_3 \cdot t\_3\right) - \log y \cdot t\_3} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fabs (log y)))
        (t_2 (* (log c) (- b 0.5)))
        (t_3 (+ (/ a x) (+ (/ t x) (+ (/ z x) (/ t_2 x)))))
        (t_4
         (*
          (* -1.0 i)
          (fma
           (*
            x
            (fma
             -1.0
             (/ (fma -1.0 (/ a i) (* -1.0 (/ (+ t (+ z t_2)) i))) x)
             (/ (log y) i)))
           -1.0
           (* -1.0 y)))))
   (if (<= i -4.8e-51)
     t_4
     (if (<= i 5.8e-73)
       (*
        (/
         (fma (pow (log y) 2.0) (log y) (pow t_3 3.0))
         (- (fma t_1 t_1 (* t_3 t_3)) (* (log y) t_3)))
        x)
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fabs(log(y));
	double t_2 = log(c) * (b - 0.5);
	double t_3 = (a / x) + ((t / x) + ((z / x) + (t_2 / x)));
	double t_4 = (-1.0 * i) * fma((x * fma(-1.0, (fma(-1.0, (a / i), (-1.0 * ((t + (z + t_2)) / i))) / x), (log(y) / i))), -1.0, (-1.0 * y));
	double tmp;
	if (i <= -4.8e-51) {
		tmp = t_4;
	} else if (i <= 5.8e-73) {
		tmp = (fma(pow(log(y), 2.0), log(y), pow(t_3, 3.0)) / (fma(t_1, t_1, (t_3 * t_3)) - (log(y) * t_3))) * x;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = abs(log(y))
	t_2 = Float64(log(c) * Float64(b - 0.5))
	t_3 = Float64(Float64(a / x) + Float64(Float64(t / x) + Float64(Float64(z / x) + Float64(t_2 / x))))
	t_4 = Float64(Float64(-1.0 * i) * fma(Float64(x * fma(-1.0, Float64(fma(-1.0, Float64(a / i), Float64(-1.0 * Float64(Float64(t + Float64(z + t_2)) / i))) / x), Float64(log(y) / i))), -1.0, Float64(-1.0 * y)))
	tmp = 0.0
	if (i <= -4.8e-51)
		tmp = t_4;
	elseif (i <= 5.8e-73)
		tmp = Float64(Float64(fma((log(y) ^ 2.0), log(y), (t_3 ^ 3.0)) / Float64(fma(t_1, t_1, Float64(t_3 * t_3)) - Float64(log(y) * t_3))) * x);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[Abs[N[Log[y], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a / x), $MachinePrecision] + N[(N[(t / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 * i), $MachinePrecision] * N[(N[(x * N[(-1.0 * N[(N[(-1.0 * N[(a / i), $MachinePrecision] + N[(-1.0 * N[(N[(t + N[(z + t$95$2), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -4.8e-51], t$95$4, If[LessEqual[i, 5.8e-73], N[(N[(N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if i < -4.8e-51 or 5.8e-73 < i

   1. Initial program 99.7%

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

   2. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Applied rewrites 84.2%

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

   if -4.8e-51 < i < 5.8e-73

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.1%

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

   6. Applied rewrites 36.5%

      \[\leadsto \frac{{\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right)}^{3} + {\log y}^{3}}{\mathsf{fma}\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \log y \cdot \log y - \left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) \cdot \log y\right)} \cdot x \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{{\log y}^{3} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{3}}{\left({\log y}^{2} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{2}\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.8%

      \[\leadsto \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\left|\log y\right|, \left|\log y\right|, \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right) \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot \left(b - 0.5\right)\\ t_2 := \frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{t\_1}{x}\right)\right)\\ t_3 := \left(-1 \cdot i\right) \cdot \mathsf{fma}\left(-1 \cdot \left(z \cdot \left(\mathsf{fma}\left(-1, \frac{a}{i \cdot z}, -1 \cdot \left(\frac{t}{i \cdot z} + \frac{\mathsf{fma}\left(x, \log y, t\_1\right)}{i \cdot z}\right)\right) - {i}^{-1}\right)\right), -1, -1 \cdot y\right)\\ t_4 := \left|\log y\right|\\ \mathbf{if}\;i \leq -4.4 \cdot 10^{-168}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;i \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\log y}^{2}, \log y, {t\_2}^{3}\right)}{\mathsf{fma}\left(t\_4, t\_4, t\_2 \cdot t\_2\right) - \log y \cdot t\_2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) (- b 0.5)))
        (t_2 (+ (/ a x) (+ (/ t x) (+ (/ z x) (/ t_1 x)))))
        (t_3
         (*
          (* -1.0 i)
          (fma
           (*
            -1.0
            (*
             z
             (-
              (fma
               -1.0
               (/ a (* i z))
               (* -1.0 (+ (/ t (* i z)) (/ (fma x (log y) t_1) (* i z)))))
              (pow i -1.0))))
           -1.0
           (* -1.0 y))))
        (t_4 (fabs (log y))))
   (if (<= i -4.4e-168)
     t_3
     (if (<= i 5e-74)
       (*
        (/
         (fma (pow (log y) 2.0) (log y) (pow t_2 3.0))
         (- (fma t_4 t_4 (* t_2 t_2)) (* (log y) t_2)))
        x)
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * (b - 0.5);
	double t_2 = (a / x) + ((t / x) + ((z / x) + (t_1 / x)));
	double t_3 = (-1.0 * i) * fma((-1.0 * (z * (fma(-1.0, (a / (i * z)), (-1.0 * ((t / (i * z)) + (fma(x, log(y), t_1) / (i * z))))) - pow(i, -1.0)))), -1.0, (-1.0 * y));
	double t_4 = fabs(log(y));
	double tmp;
	if (i <= -4.4e-168) {
		tmp = t_3;
	} else if (i <= 5e-74) {
		tmp = (fma(pow(log(y), 2.0), log(y), pow(t_2, 3.0)) / (fma(t_4, t_4, (t_2 * t_2)) - (log(y) * t_2))) * x;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * Float64(b - 0.5))
	t_2 = Float64(Float64(a / x) + Float64(Float64(t / x) + Float64(Float64(z / x) + Float64(t_1 / x))))
	t_3 = Float64(Float64(-1.0 * i) * fma(Float64(-1.0 * Float64(z * Float64(fma(-1.0, Float64(a / Float64(i * z)), Float64(-1.0 * Float64(Float64(t / Float64(i * z)) + Float64(fma(x, log(y), t_1) / Float64(i * z))))) - (i ^ -1.0)))), -1.0, Float64(-1.0 * y)))
	t_4 = abs(log(y))
	tmp = 0.0
	if (i <= -4.4e-168)
		tmp = t_3;
	elseif (i <= 5e-74)
		tmp = Float64(Float64(fma((log(y) ^ 2.0), log(y), (t_2 ^ 3.0)) / Float64(fma(t_4, t_4, Float64(t_2 * t_2)) - Float64(log(y) * t_2))) * x);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / x), $MachinePrecision] + N[(N[(t / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 * i), $MachinePrecision] * N[(N[(-1.0 * N[(z * N[(N[(-1.0 * N[(a / N[(i * z), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(t / N[(i * z), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision] + t$95$1), $MachinePrecision] / N[(i * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[i, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[Log[y], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[i, -4.4e-168], t$95$3, If[LessEqual[i, 5e-74], N[(N[(N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 * t$95$4 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if i < -4.3999999999999996e-168 or 4.99999999999999998e-74 < i

   1. Initial program 99.7%

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

   2. Taylor expanded in i around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in z around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 71.6%

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

   if -4.3999999999999996e-168 < i < 4.99999999999999998e-74

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   6. Applied rewrites 36.2%

      \[\leadsto \frac{{\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right)}^{3} + {\log y}^{3}}{\mathsf{fma}\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \log y \cdot \log y - \left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) \cdot \log y\right)} \cdot x \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{{\log y}^{3} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{3}}{\left({\log y}^{2} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{2}\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)} \cdot x \]
   8. Step-by-step derivation

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\left|\log y\right|, \left|\log y\right|, \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right) \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 28.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\\ t_2 := \left|\log y\right|\\ \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {t\_1}^{3}\right)}{\mathsf{fma}\left(t\_2, t\_2, t\_1 \cdot t\_1\right) - \log y \cdot t\_1} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ a x) (+ (/ t x) (+ (/ z x) (/ (* (log c) (- b 0.5)) x)))))
        (t_2 (fabs (log y))))
   (*
    (/
     (fma (pow (log y) 2.0) (log y) (pow t_1 3.0))
     (- (fma t_2 t_2 (* t_1 t_1)) (* (log y) t_1)))
    x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a / x) + ((t / x) + ((z / x) + ((log(c) * (b - 0.5)) / x)));
	double t_2 = fabs(log(y));
	return (fma(pow(log(y), 2.0), log(y), pow(t_1, 3.0)) / (fma(t_2, t_2, (t_1 * t_1)) - (log(y) * t_1))) * x;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a / x) + Float64(Float64(t / x) + Float64(Float64(z / x) + Float64(Float64(log(c) * Float64(b - 0.5)) / x))))
	t_2 = abs(log(y))
	return Float64(Float64(fma((log(y) ^ 2.0), log(y), (t_1 ^ 3.0)) / Float64(fma(t_2, t_2, Float64(t_1 * t_1)) - Float64(log(y) * t_1))) * x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a / x), $MachinePrecision] + N[(N[(t / x), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] + N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[Log[y], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision] * N[Log[y], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * t$95$2 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 70.4%

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

6. Applied rewrites 29.8%

   \[\leadsto \frac{{\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right)}^{3} + {\log y}^{3}}{\mathsf{fma}\left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}, \log y \cdot \log y - \left(\left(\frac{t + z}{x} + \mathsf{fma}\left(\log c, \frac{b - 0.5}{x}, \frac{i \cdot y}{x}\right)\right) + \frac{a}{x}\right) \cdot \log y\right)} \cdot x \]

7. Taylor expanded in y around 0

   \[\leadsto \frac{{\log y}^{3} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{3}}{\left({\log y}^{2} + {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)}^{2}\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{x}\right)\right)\right)} \cdot x \]
8. Step-by-step derivation

9. Applied rewrites 28.4%

   \[\leadsto \frac{\mathsf{fma}\left({\log y}^{2}, \log y, {\left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\left|\log y\right|, \left|\log y\right|, \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right) \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)\right) - \log y \cdot \left(\frac{a}{x} + \left(\frac{t}{x} + \left(\frac{z}{x} + \frac{\log c \cdot \left(b - 0.5\right)}{x}\right)\right)\right)} \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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)))