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

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 20

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 99.8%

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

Alternative 2: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+253}:\\ \;\;\;\;z + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+61}:\\ \;\;\;\;\left(t + z\right) + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

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 -2e+253)
       (+ z (* (log c) b))
       (if (<= t_1 -4e+61)
         (+ (+ t z) (* (log y) x))
         (fma y i (fma (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 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 <= -2e+253) {
		tmp = z + (log(c) * b);
	} else if (t_1 <= -4e+61) {
		tmp = (t + z) + (log(y) * x);
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), a));
	}
	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 <= -2e+253)
		tmp = Float64(z + Float64(log(c) * b));
	elseif (t_1 <= -4e+61)
		tmp = Float64(Float64(t + z) + Float64(log(y) * x));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), a));
	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, -2e+253], N[(z + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e+61], N[(N[(t + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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

   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. Taylor expanded in y around inf

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

      1. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   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)) < -1.9999999999999999e253

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 54.7

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   7. Applied rewrites 54.7%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   8. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log c} \cdot b \]
   9. Step-by-step derivation

   10. Applied rewrites 35.6%

       \[\leadsto z + \color{blue}{\log c} \cdot b \]

   if -1.9999999999999999e253 < (+.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)) < -3.9999999999999998e61

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(t + z\right) + x \cdot \color{blue}{\log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      3. lift-*.f64 52.6

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

   7. Applied rewrites 52.6%

      \[\leadsto \left(t + z\right) + \log y \cdot \color{blue}{x} \]

   if -3.9999999999999998e61 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.9

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

   4. Applied rewrites 88.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.9

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.9%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 54.0%

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

Alternative 3: 44.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+253}:\\ \;\;\;\;z + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\left(t + z\right) + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -2e+253)
       (+ z (* (log c) b))
       (if (<= t_1 -5e+33) (+ (+ t z) (* (log y) x)) (fma y i a))))))

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 <= -2e+253) {
		tmp = z + (log(c) * b);
	} else if (t_1 <= -5e+33) {
		tmp = (t + z) + (log(y) * x);
	} else {
		tmp = fma(y, i, a);
	}
	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 <= -2e+253)
		tmp = Float64(z + Float64(log(c) * b));
	elseif (t_1 <= -5e+33)
		tmp = Float64(Float64(t + z) + Float64(log(y) * x));
	else
		tmp = fma(y, i, a);
	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, -2e+253], N[(z + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+33], N[(N[(t + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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

   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. Taylor expanded in y around inf

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

      1. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   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)) < -1.9999999999999999e253

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 54.7

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   7. Applied rewrites 54.7%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   8. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log c} \cdot b \]
   9. Step-by-step derivation

   10. Applied rewrites 35.6%

       \[\leadsto z + \color{blue}{\log c} \cdot b \]

   if -1.9999999999999999e253 < (+.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)) < -4.99999999999999973e33

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(t + z\right) + x \cdot \color{blue}{\log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      3. lift-*.f64 52.4

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

   7. Applied rewrites 52.4%

      \[\leadsto \left(t + z\right) + \log y \cdot \color{blue}{x} \]

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 45.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+205}:\\ \;\;\;\;\left(t + z\right) + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;z + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -2e+205)
       (+ (+ t z) (* (log c) b))
       (if (<= t_1 -5e+33) (+ z (* (log y) x)) (fma y i a))))))

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 <= -2e+205) {
		tmp = (t + z) + (log(c) * b);
	} else if (t_1 <= -5e+33) {
		tmp = z + (log(y) * x);
	} else {
		tmp = fma(y, i, a);
	}
	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 <= -2e+205)
		tmp = Float64(Float64(t + z) + Float64(log(c) * b));
	elseif (t_1 <= -5e+33)
		tmp = Float64(z + Float64(log(y) * x));
	else
		tmp = fma(y, i, a);
	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, -2e+205], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+33], N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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

   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. Taylor expanded in y around inf

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

      1. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   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)) < -2.00000000000000003e205

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 54.7

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   7. Applied rewrites 54.7%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   if -2.00000000000000003e205 < (+.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)) < -4.99999999999999973e33

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(t + z\right) + x \cdot \color{blue}{\log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      3. lift-*.f64 52.3

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

   7. Applied rewrites 52.3%

      \[\leadsto \left(t + z\right) + \log y \cdot \color{blue}{x} \]

   8. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 34.9%

       \[\leadsto z + \color{blue}{\log y} \cdot x \]

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 40.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+253}:\\ \;\;\;\;z + \log c \cdot b\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;z + \log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -2e+253)
       (+ z (* (log c) b))
       (if (<= t_1 -5e+33) (+ z (* (log y) x)) (fma y i a))))))

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 <= -2e+253) {
		tmp = z + (log(c) * b);
	} else if (t_1 <= -5e+33) {
		tmp = z + (log(y) * x);
	} else {
		tmp = fma(y, i, a);
	}
	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 <= -2e+253)
		tmp = Float64(z + Float64(log(c) * b));
	elseif (t_1 <= -5e+33)
		tmp = Float64(z + Float64(log(y) * x));
	else
		tmp = fma(y, i, a);
	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, -2e+253], N[(z + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+33], N[(z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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

   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. Taylor expanded in y around inf

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

      1. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   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)) < -1.9999999999999999e253

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 54.7

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   7. Applied rewrites 54.7%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   8. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log c} \cdot b \]
   9. Step-by-step derivation

   10. Applied rewrites 35.6%

       \[\leadsto z + \color{blue}{\log c} \cdot b \]

   if -1.9999999999999999e253 < (+.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)) < -4.99999999999999973e33

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(t + z\right) + x \cdot \color{blue}{\log y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

      3. lift-*.f64 52.4

         \[\leadsto \left(t + z\right) + \log y \cdot x \]

   7. Applied rewrites 52.4%

      \[\leadsto \left(t + z\right) + \log y \cdot \color{blue}{x} \]

   8. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log y} \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 34.1%

       \[\leadsto z + \color{blue}{\log y} \cdot x \]

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 28.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;z\\ \mathbf{elif}\;t\_1 \leq 10^{+296}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

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 -100.0) z (if (<= t_1 1e+296) a (* 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 <= -100.0) {
		tmp = z;
	} else if (t_1 <= 1e+296) {
		tmp = a;
	} else {
		tmp = i * y;
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = i * y;
	} else if (t_1 <= -100.0) {
		tmp = z;
	} else if (t_1 <= 1e+296) {
		tmp = a;
	} 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 <= -math.inf:
		tmp = i * y
	elif t_1 <= -100.0:
		tmp = z
	elif t_1 <= 1e+296:
		tmp = a
	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 <= -100.0)
		tmp = z;
	elseif (t_1 <= 1e+296)
		tmp = a;
	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 <= -Inf)
		tmp = i * y;
	elseif (t_1 <= -100.0)
		tmp = z;
	elseif (t_1 <= 1e+296)
		tmp = a;
	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, (-Infinity)], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -100.0], z, If[LessEqual[t$95$1, 1e+296], a, 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 9.99999999999999981e295 < (+.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 99.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 73.5

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

   4. Applied rewrites 73.5%

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

   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)) < -100

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 18.0%

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

   if -100 < (+.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)) < 9.99999999999999981e295

   1. Initial program 99.8%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 17.7%

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

Alternative 7: 40.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;z + \log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -5e+33) (+ z (* (log c) b)) (fma y i a)))))

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 <= -5e+33) {
		tmp = z + (log(c) * b);
	} else {
		tmp = fma(y, i, a);
	}
	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 <= -5e+33)
		tmp = Float64(z + Float64(log(c) * b));
	else
		tmp = fma(y, i, a);
	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, -5e+33], N[(z + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y * i + a), $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

   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. Taylor expanded in y around inf

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

      1. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   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)) < -4.99999999999999973e33

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in b around inf

      \[\leadsto \left(t + z\right) + b \cdot \color{blue}{\log c} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      2. lift-log.f64 N/A

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

      3. lift-*.f64 53.0

         \[\leadsto \left(t + z\right) + \log c \cdot b \]

   7. Applied rewrites 53.0%

      \[\leadsto \left(t + z\right) + \log c \cdot \color{blue}{b} \]

   8. Taylor expanded in z around inf

      \[\leadsto z + \color{blue}{\log c} \cdot b \]
   9. Step-by-step derivation

   10. Applied rewrites 34.5%

       \[\leadsto z + \color{blue}{\log c} \cdot b \]

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 33.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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 -5e+33) z (fma y i a)))))

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 <= -5e+33) {
		tmp = z;
	} else {
		tmp = fma(y, i, a);
	}
	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 <= -5e+33)
		tmp = z;
	else
		tmp = fma(y, i, a);
	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, -5e+33], z, N[(y * i + a), $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

   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. Taylor expanded in y around inf

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

      1. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   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)) < -4.99999999999999973e33

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 18.3%

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

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, t + a\right)\right)\\ \end{array} \end{array} \]

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))
      -5e+33)
   (fma y i (fma (log c) (- b 0.5) z))
   (fma y i (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)) <= -5e+33) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), z));
	} else {
		tmp = fma(y, i, 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)) <= -5e+33)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), z));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(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], -5e+33], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision]), $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)) < -4.99999999999999973e33

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 54.0%

      \[\leadsto \left(\color{blue}{z} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 54.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 54.0

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

   6. Applied rewrites 54.0%

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

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 69.0%

      \[\leadsto \left(\left(\color{blue}{t} + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 69.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 69.0

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

   6. Applied rewrites 69.0%

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

Alternative 10: 54.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, a\right)\right)\\ \end{array} \end{array} \]

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))
      -5e+33)
   (fma y i (fma (log c) (- b 0.5) z))
   (fma y i (fma (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)) <= -5e+33) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), z));
	} else {
		tmp = fma(y, i, fma(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)) <= -5e+33)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), z));
	else
		tmp = fma(y, i, fma(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], -5e+33], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + a), $MachinePrecision]), $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)) < -4.99999999999999973e33

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 54.0%

      \[\leadsto \left(\color{blue}{z} + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 54.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. lift--.f64 54.0

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

   6. Applied rewrites 54.0%

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

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 54.0%

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

Alternative 11: 45.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\left(t + z\right) + i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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))
      -5e+33)
   (+ (+ t z) (* i y))
   (fma y i 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)) <= -5e+33) {
		tmp = (t + z) + (i * y);
	} else {
		tmp = fma(y, i, 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)) <= -5e+33)
		tmp = Float64(Float64(t + z) + Float64(i * y));
	else
		tmp = fma(y, i, 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], -5e+33], N[(N[(t + z), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision], N[(y * i + 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)) < -4.99999999999999973e33

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 84.7

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto \left(t + z\right) + i \cdot \color{blue}{y} \]
   6. Step-by-step derivation

      1. lower-*.f64 55.2

         \[\leadsto \left(t + z\right) + i \cdot y \]

   7. Applied rewrites 55.2%

      \[\leadsto \left(t + z\right) + i \cdot \color{blue}{y} \]

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

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))
      -5e+33)
   (fma y i z)
   (fma y i 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)) <= -5e+33) {
		tmp = fma(y, i, z);
	} else {
		tmp = fma(y, i, 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)) <= -5e+33)
		tmp = fma(y, i, z);
	else
		tmp = fma(y, i, 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], -5e+33], N[(y * i + z), $MachinePrecision], N[(y * i + 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)) < -4.99999999999999973e33

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 89.2

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 89.3%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{z}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 39.6%

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

   if -4.99999999999999973e33 < (+.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 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.7

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 88.7%

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

   7. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 15.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -100:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

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))
      -100.0)
   z
   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)) <= -100.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, 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 (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-100.0d0)) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -100.0) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -100.0:
		tmp = z
	else:
		tmp = 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)) <= -100.0)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -100.0)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = 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], -100.0], z, a]

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)) < -100

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 15.7%

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

   if -100 < (+.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 99.7%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 15.8%

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

Alternative 14: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + t\right) + \left(\mathsf{fma}\left(\log y, x, z\right) + \log c \cdot \left(b - 0.5\right)\right)\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ a t) (+ (fma (log y) x z) (* (log c) (- b 0.5))))))
   (if (<= x -6.4e+112)
     t_1
     (if (<= x 3.8e+160)
       (fma y i (fma (log c) (- b 0.5) (+ z (+ t a))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + t) + (fma(log(y), x, z) + (log(c) * (b - 0.5)));
	double tmp;
	if (x <= -6.4e+112) {
		tmp = t_1;
	} else if (x <= 3.8e+160) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (z + (t + a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + t) + Float64(fma(log(y), x, z) + Float64(log(c) * Float64(b - 0.5))))
	tmp = 0.0
	if (x <= -6.4e+112)
		tmp = t_1;
	elseif (x <= 3.8e+160)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(z + Float64(t + a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + t), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.4e+112], t$95$1, If[LessEqual[x, 3.8e+160], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(z + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.39999999999999972e112 or 3.80000000000000012e160 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 80.9

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

   4. Applied rewrites 80.9%

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

   if -6.39999999999999972e112 < x < 3.80000000000000012e160

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 96.3

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 96.8%

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

Alternative 15: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+223}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (fma (log c) (- b 0.5) (* (log y) x)))))
   (if (<= x -4e+187)
     t_1
     (if (<= x 6.6e+223)
       (fma y i (fma (log c) (- b 0.5) (+ z (+ t a))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, fma(log(c), (b - 0.5), (log(y) * x)));
	double tmp;
	if (x <= -4e+187) {
		tmp = t_1;
	} else if (x <= 6.6e+223) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (z + (t + a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(log(y) * x)))
	tmp = 0.0
	if (x <= -4e+187)
		tmp = t_1;
	elseif (x <= 6.6e+223)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(z + Float64(t + a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+187], t$95$1, If[LessEqual[x, 6.6e+223], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(z + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999963e187 or 6.5999999999999999e223 < 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. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 66.2

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 66.2%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 81.4

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

   9. Applied rewrites 81.4%

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

   if -3.99999999999999963e187 < x < 6.5999999999999999e223

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 93.2

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

   4. Applied rewrites 93.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.2

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 93.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 93.1%

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

Alternative 16: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(y, i, t\_1\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + z\right) + \mathsf{fma}\left(\log c, b - 0.5, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -6.6e+189)
     (fma y i t_1)
     (if (<= x 2.8e+198)
       (fma y i (fma (log c) (- b 0.5) (+ z (+ t a))))
       (+ (+ t z) (fma (log c) (- b 0.5) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -6.6e+189) {
		tmp = fma(y, i, t_1);
	} else if (x <= 2.8e+198) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (z + (t + a))));
	} else {
		tmp = (t + z) + fma(log(c), (b - 0.5), t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -6.6e+189)
		tmp = fma(y, i, t_1);
	elseif (x <= 2.8e+198)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(z + Float64(t + a))));
	else
		tmp = Float64(Float64(t + z) + fma(log(c), Float64(b - 0.5), t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.6e+189], N[(y * i + t$95$1), $MachinePrecision], If[LessEqual[x, 2.8e+198], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(z + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6000000000000004e189

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 66.1

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

   4. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 66.1

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 66.1%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{x \cdot \log y}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, \log y \cdot \color{blue}{x}\right) \]

      2. lift-log.f64 N/A

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

      3. lift-*.f64 73.3

         \[\leadsto \mathsf{fma}\left(y, i, \log y \cdot \color{blue}{x}\right) \]

   9. Applied rewrites 73.3%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log y \cdot x}\right) \]

   if -6.6000000000000004e189 < x < 2.8e198

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 93.9

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

   4. Applied rewrites 93.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.9

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 93.9%

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

   if 2.8e198 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 77.5

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

   7. Applied rewrites 77.5%

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

Alternative 17: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+155}:\\ \;\;\;\;\left(t + z\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z + \left(t + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2.7e+155)
   (+ (+ t z) (fma i y (fma (log c) (- b 0.5) (* (log y) x))))
   (fma y i (fma (log c) (- b 0.5) (+ z (+ t a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2.7e+155) {
		tmp = (t + z) + fma(i, y, fma(log(c), (b - 0.5), (log(y) * x)));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (z + (t + a))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2.7e+155)
		tmp = Float64(Float64(t + z) + fma(i, y, fma(log(c), Float64(b - 0.5), Float64(log(y) * x))));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(z + Float64(t + a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2.7e+155], N[(N[(t + z), $MachinePrecision] + N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(z + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.69999999999999994e155

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 89.7

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

   4. Applied rewrites 89.7%

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

   if 2.69999999999999994e155 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 87.6

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 91.8%

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

Alternative 18: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+155}:\\ \;\;\;\;\left(t + z\right) + \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \log c \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z + \left(t + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2.7e+155)
   (+ (+ t z) (fma i y (fma (log y) x (* (log c) b))))
   (fma y i (fma (log c) (- b 0.5) (+ z (+ t a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2.7e+155) {
		tmp = (t + z) + fma(i, y, fma(log(y), x, (log(c) * b)));
	} else {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (z + (t + a))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2.7e+155)
		tmp = Float64(Float64(t + z) + fma(i, y, fma(log(y), x, Float64(log(c) * b))));
	else
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(z + Float64(t + a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2.7e+155], N[(N[(t + z), $MachinePrecision] + N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(z + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.69999999999999994e155

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-log.f64 89.7

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

   4. Applied rewrites 89.7%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 89.8

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

   6. Applied rewrites 89.8%

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

   7. Taylor expanded in b around inf

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

   9. Applied rewrites 88.0%

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

   if 2.69999999999999994e155 < a

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 87.6

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 87.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 91.8%

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

Alternative 19: 90.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, \log y \cdot x\right)\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+231}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(\log c, b - 0.5, z + \left(t + a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* (log y) x))))
   (if (<= x -6.6e+189)
     t_1
     (if (<= x 1.3e+231)
       (fma y i (fma (log c) (- b 0.5) (+ z (+ t a))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, i, (log(y) * x));
	double tmp;
	if (x <= -6.6e+189) {
		tmp = t_1;
	} else if (x <= 1.3e+231) {
		tmp = fma(y, i, fma(log(c), (b - 0.5), (z + (t + a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(log(y) * x))
	tmp = 0.0
	if (x <= -6.6e+189)
		tmp = t_1;
	elseif (x <= 1.3e+231)
		tmp = fma(y, i, fma(log(c), Float64(b - 0.5), Float64(z + Float64(t + a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * i + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.6e+189], t$95$1, If[LessEqual[x, 1.3e+231], N[(y * i + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(z + N[(t + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6000000000000004e189 or 1.2999999999999999e231 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 66.0

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

   4. Applied rewrites 66.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 66.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 66.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{x \cdot \log y}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, i, \log y \cdot \color{blue}{x}\right) \]

      2. lift-log.f64 N/A

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

      3. lift-*.f64 75.8

         \[\leadsto \mathsf{fma}\left(y, i, \log y \cdot \color{blue}{x}\right) \]

   9. Applied rewrites 75.8%

      \[\leadsto \mathsf{fma}\left(y, i, \color{blue}{\log y \cdot x}\right) \]

   if -6.6000000000000004e189 < x < 1.2999999999999999e231

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 93.0

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

   4. Applied rewrites 93.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.0

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 93.0%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 92.8%

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

Alternative 20: 15.9% accurate, 234.0× speedup

Math

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

FPCore

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

C

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

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 = a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in a around inf

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

4. Applied rewrites 15.9%

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

Reproduce

herbie shell --seed 2025095 
(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)))