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

Percentage Accurate: 99.8% → 99.8%

Time: 10.8s

Alternatives: 15

Speedup: 1.0×

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, 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]

Derivation

1. Initial program 99.9%

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

Alternative 2: 30.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, z, z\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 (- INFINITY))
     (* i y)
     (if (<= t_1 -200.0)
       (fma (/ t z) z z)
       (if (<= t_1 2e+306) (* (/ a i) i) (* i y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = i * y;
	} else if (t_1 <= -200.0) {
		tmp = fma((t / z), z, z);
	} else if (t_1 <= 2e+306) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(i * y);
	elseif (t_1 <= -200.0)
		tmp = fma(Float64(t / z), z, z);
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(a / i) * i);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -200.0], N[(N[(t / z), $MachinePrecision] * z + z), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision], N[(i * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -inf.0 or 2.00000000000000003e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 93.2

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -inf.0 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -200

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.9%

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

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{fma}\left(\frac{t}{z}, z, z\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 28.4%

       \[\leadsto \mathsf{fma}\left(\frac{t}{z}, z, z\right) \]

   if -200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2.00000000000000003e306

   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. Add Preprocessing

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 13.1%

      \[\leadsto \frac{a}{i} \cdot i \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 23.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+306)
     (* i y)
     (if (<= t_1 -200.0)
       (* (/ z i) i)
       (if (<= t_1 2e+306) (* (/ a i) i) (* i y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = i * y;
	} else if (t_1 <= -200.0) {
		tmp = (z / i) * i;
	} else if (t_1 <= 2e+306) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, 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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= (-2d+306)) then
        tmp = i * y
    else if (t_1 <= (-200.0d0)) then
        tmp = (z / i) * i
    else if (t_1 <= 2d+306) then
        tmp = (a / i) * i
    else
        tmp = i * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = i * y;
	} else if (t_1 <= -200.0) {
		tmp = (z / i) * i;
	} else if (t_1 <= 2e+306) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -2e+306:
		tmp = i * y
	elif t_1 <= -200.0:
		tmp = (z / i) * i
	elif t_1 <= 2e+306:
		tmp = (a / i) * i
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(i * y);
	elseif (t_1 <= -200.0)
		tmp = Float64(Float64(z / i) * i);
	elseif (t_1 <= 2e+306)
		tmp = Float64(Float64(a / i) * i);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -2e+306)
		tmp = i * y;
	elseif (t_1 <= -200.0)
		tmp = (z / i) * i;
	elseif (t_1 <= 2e+306)
		tmp = (a / i) * i;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e+306], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -200.0], N[(N[(z / i), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision], N[(i * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000003e306 or 2.00000000000000003e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 87.6

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 87.6%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -2.00000000000000003e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -200

   1. Initial program 99.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{z}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 9.0%

      \[\leadsto \frac{z}{i} \cdot i \]

   if -200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2.00000000000000003e306

   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. Add Preprocessing

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 13.1%

      \[\leadsto \frac{a}{i} \cdot i \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 29.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -200 \lor \neg \left(t\_1 \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (or (<= t_1 -200.0) (not (<= t_1 2e+306)))
     (fma (/ (* i y) z) z z)
     (* (/ a i) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if ((t_1 <= -200.0) || !(t_1 <= 2e+306)) {
		tmp = fma(((i * y) / z), z, z);
	} else {
		tmp = (a / i) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if ((t_1 <= -200.0) || !(t_1 <= 2e+306))
		tmp = fma(Float64(Float64(i * y) / z), z, z);
	else
		tmp = Float64(Float64(a / i) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, -200.0], N[Not[LessEqual[t$95$1, 2e+306]], $MachinePrecision]], N[(N[(N[(i * y), $MachinePrecision] / z), $MachinePrecision] * z + z), $MachinePrecision], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -200 or 2.00000000000000003e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 78.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 66.9%

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

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 39.9%

       \[\leadsto \mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right) \]

   if -200 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 2.00000000000000003e306

   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. Add Preprocessing

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 13.1%

      \[\leadsto \frac{a}{i} \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -200 \lor \neg \left(\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 2 \cdot 10^{+306}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, t\_1\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log y) x (fma (- b 0.5) (log c) t))))
   (if (<=
        (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
        -1000000.0)
     (+ (fma i y z) t_1)
     (+ (fma i y t_1) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(y), x, fma((b - 0.5), log(c), t));
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -1000000.0) {
		tmp = fma(i, y, z) + t_1;
	} else {
		tmp = fma(i, y, t_1) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(y), x, fma(Float64(b - 0.5), log(c), t))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -1000000.0)
		tmp = Float64(fma(i, y, z) + t_1);
	else
		tmp = Float64(fma(i, y, t_1) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -1000000.0], N[(N[(i * y + z), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(i * y + t$95$1), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -1e6 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.1%

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

Alternative 6: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -1000000.0)
   (+ (fma i y z) (fma (log y) x (* 1.0 t)))
   (+ (fma i y (fma (log c) (- b 0.5) t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -1000000.0) {
		tmp = fma(i, y, z) + fma(log(y), x, (1.0 * t));
	} else {
		tmp = fma(i, y, fma(log(c), (b - 0.5), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -1000000.0)
		tmp = Float64(fma(i, y, z) + fma(log(y), x, Float64(1.0 * t)));
	else
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], -1000000.0], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(1.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 73.7%

       \[\leadsto \mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right) \]

   if -1e6 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

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

Alternative 7: 70.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log c, b - 0.5, t\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, t\_1\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log c) (- b 0.5) t)))
   (if (<=
        (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
        -1000000.0)
     (+ (fma i y z) t_1)
     (+ (fma i y t_1) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(c), (b - 0.5), t);
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -1000000.0) {
		tmp = fma(i, y, z) + t_1;
	} else {
		tmp = fma(i, y, t_1) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(c), Float64(b - 0.5), t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -1000000.0)
		tmp = Float64(fma(i, y, z) + t_1);
	else
		tmp = Float64(fma(i, y, t_1) + a);
	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] + t), $MachinePrecision]}, If[LessEqual[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], -1000000.0], N[(N[(i * y + z), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(i * y + t$95$1), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.8%

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

   if -1e6 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

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

Alternative 8: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b - 0.5, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -1000000.0)
   (+ (fma i y z) (fma (log c) (- b 0.5) t))
   (+ (fma i y (* (log c) (- b 0.5))) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -1000000.0) {
		tmp = fma(i, y, z) + fma(log(c), (b - 0.5), t);
	} else {
		tmp = fma(i, y, (log(c) * (b - 0.5))) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -1000000.0)
		tmp = Float64(fma(i, y, z) + fma(log(c), Float64(b - 0.5), t));
	else
		tmp = Float64(fma(i, y, Float64(log(c) * Float64(b - 0.5))) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], -1000000.0], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.8%

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

   if -1e6 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 55.4%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log c, b - 0.5, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(-0.5, \log c, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
      -1000000.0)
   (fma (/ (* i y) z) z z)
   (+ (fma i y (fma -0.5 (log c) t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -1000000.0) {
		tmp = fma(((i * y) / z), z, z);
	} else {
		tmp = fma(i, y, fma(-0.5, log(c), t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -1000000.0)
		tmp = fma(Float64(Float64(i * y) / z), z, z);
	else
		tmp = Float64(fma(i, y, fma(-0.5, log(c), t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], -1000000.0], N[(N[(N[(i * y), $MachinePrecision] / z), $MachinePrecision] * z + z), $MachinePrecision], N[(N[(i * y + N[(-0.5 * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -1e6

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 63.5%

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

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 33.4%

       \[\leadsto \mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right) \]

   if -1e6 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 54.7%

       \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(-0.5, \log c, t\right)\right) + a \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -1000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(-0.5, \log c, t\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+229}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log y, x, z\right) + t\right) + \log c \cdot \left(b - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x -8.2e+217)
   (+ (fma i y z) (fma (log y) x (* 1.0 t)))
   (if (<= x 3.9e+229)
     (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))
     (+ (+ (fma (log y) x z) t) (* (log c) (- b 0.5))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= -8.2e+217) {
		tmp = fma(i, y, z) + fma(log(y), x, (1.0 * t));
	} else if (x <= 3.9e+229) {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	} else {
		tmp = (fma(log(y), x, z) + t) + (log(c) * (b - 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= -8.2e+217)
		tmp = Float64(fma(i, y, z) + fma(log(y), x, Float64(1.0 * t)));
	elseif (x <= 3.9e+229)
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	else
		tmp = Float64(Float64(fma(log(y), x, z) + t) + Float64(log(c) * Float64(b - 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, -8.2e+217], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(1.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e+229], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision] + t), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000005e217

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.2%

       \[\leadsto \mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right) \]

   if -8.2000000000000005e217 < x < 3.8999999999999998e229

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 3.8999999999999998e229 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 93.7%

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

Alternative 11: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2e+53)
   (+ (fma i y z) (fma (log y) x (fma (- b 0.5) (log c) t)))
   (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2e+53) {
		tmp = fma(i, y, z) + fma(log(y), x, fma((b - 0.5), log(c), t));
	} else {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2e+53)
		tmp = Float64(fma(i, y, z) + fma(log(y), x, fma(Float64(b - 0.5), log(c), t)));
	else
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2e+53], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2e53

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if 2e53 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 94.1

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

   5. Applied rewrites 94.1%

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

Alternative 12: 91.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+217} \lor \neg \left(x \leq 3.05 \cdot 10^{+88}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= x -8.2e+217) (not (<= x 3.05e+88)))
   (+ (fma i y z) (fma (log y) x (* 1.0 t)))
   (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((x <= -8.2e+217) || !(x <= 3.05e+88)) {
		tmp = fma(i, y, z) + fma(log(y), x, (1.0 * t));
	} else {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((x <= -8.2e+217) || !(x <= 3.05e+88))
		tmp = Float64(fma(i, y, z) + fma(log(y), x, Float64(1.0 * t)));
	else
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[x, -8.2e+217], N[Not[LessEqual[x, 3.05e+88]], $MachinePrecision]], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(1.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000005e217 or 3.0499999999999999e88 < x

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 83.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 87.1%

       \[\leadsto \mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right) \]

   if -8.2000000000000005e217 < x < 3.0499999999999999e88

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+217} \lor \neg \left(x \leq 3.05 \cdot 10^{+88}\right):\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, 1 \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -6.2e+137)
   (fma (/ (* i y) z) z z)
   (+ (fma i y (* (log c) (- b 0.5))) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -6.2e+137) {
		tmp = fma(((i * y) / z), z, z);
	} else {
		tmp = fma(i, y, (log(c) * (b - 0.5))) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -6.2e+137)
		tmp = fma(Float64(Float64(i * y) / z), z, z);
	else
		tmp = Float64(fma(i, y, Float64(log(c) * Float64(b - 0.5))) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -6.2e+137], N[(N[(N[(i * y), $MachinePrecision] / z), $MachinePrecision] * z + z), $MachinePrecision], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999999e137

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 70.4%

       \[\leadsto \mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right) \]

   if -6.1999999999999999e137 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 55.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{i \cdot y}{z}, z, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \log c \cdot \left(b - 0.5\right)\right) + a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.35 \cdot 10^{+184}:\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 3.35e+184) (* i y) (* (/ a i) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 3.35e+184) {
		tmp = i * y;
	} else {
		tmp = (a / i) * i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, 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
    real(8) :: tmp
    if (a <= 3.35d+184) then
        tmp = i * y
    else
        tmp = (a / i) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 3.35e+184) {
		tmp = i * y;
	} else {
		tmp = (a / i) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if a <= 3.35e+184:
		tmp = i * y
	else:
		tmp = (a / i) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 3.35e+184)
		tmp = Float64(i * y);
	else
		tmp = Float64(Float64(a / i) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (a <= 3.35e+184)
		tmp = i * y;
	else
		tmp = (a / i) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 3.35e+184], N[(i * y), $MachinePrecision], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.35e184

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 23.5

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 23.5%

      \[\leadsto \color{blue}{i \cdot y} \]

   if 3.35e184 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 32.4%

      \[\leadsto \frac{a}{i} \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 24.6% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ i \cdot y \end{array} \]

FPCore

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

C

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

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 = i * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(i * y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{i \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 22.8

      \[\leadsto \color{blue}{i \cdot y} \]

5. Applied rewrites 22.8%

   \[\leadsto \color{blue}{i \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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)))