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

Percentage Accurate: 99.7% → 99.7%

Time: 7.2s

Alternatives: 17

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: 99.7% 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: 99.7% 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 99.7%

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

Alternative 2: 92.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(t$95$1 + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], 200.0], N[(N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision], N[(y * i + N[(b * N[Log[c], $MachinePrecision] + N[(t$95$1 + 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)) < 200

   1. Initial program 99.9%

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

   2. 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 84.3

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

   4. Applied rewrites 84.3%

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

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

   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 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 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 87.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 87.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 87.5%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lift-log.f64 70.1

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

   12. Applied rewrites 70.1%

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

Alternative 3: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(i, y, x \cdot \log y\right) + z\right) + t\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\log c, b - 0.5, i \cdot y\right) + z\right) + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (fma i y (* x (log y))) z) t)))
   (if (<= x -2.05e+165)
     t_1
     (if (<= x 7.4e+158)
       (+ (+ (+ (fma (log c) (- b 0.5) (* i y)) 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(i, y, (x * log(y))) + z) + t;
	double tmp;
	if (x <= -2.05e+165) {
		tmp = t_1;
	} else if (x <= 7.4e+158) {
		tmp = ((fma(log(c), (b - 0.5), (i * y)) + z) + t) + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(i, y, Float64(x * log(y))) + z) + t)
	tmp = 0.0
	if (x <= -2.05e+165)
		tmp = t_1;
	elseif (x <= 7.4e+158)
		tmp = Float64(Float64(Float64(fma(log(c), Float64(b - 0.5), Float64(i * y)) + z) + 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[(N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[x, -2.05e+165], t$95$1, If[LessEqual[x, 7.4e+158], N[(N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e165 or 7.40000000000000021e158 < 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 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -2.0500000000000001e165 < x < 7.40000000000000021e158

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

Alternative 4: 78.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(i, y, x \cdot \log y\right) + z\right) + t\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;\left(\left(z + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (fma i y (* x (log y))) z) t)))
   (if (<= x -2.05e+165)
     t_1
     (if (<= x 7.4e+158) (+ (+ (+ z a) (* (- b 0.5) (log c))) (* y i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (fma(i, y, (x * log(y))) + z) + t;
	double tmp;
	if (x <= -2.05e+165) {
		tmp = t_1;
	} else if (x <= 7.4e+158) {
		tmp = ((z + a) + ((b - 0.5) * log(c))) + (y * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(i, y, Float64(x * log(y))) + z) + t)
	tmp = 0.0
	if (x <= -2.05e+165)
		tmp = t_1;
	elseif (x <= 7.4e+158)
		tmp = Float64(Float64(Float64(z + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[x, -2.05e+165], t$95$1, If[LessEqual[x, 7.4e+158], N[(N[(N[(z + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e165 or 7.40000000000000021e158 < 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 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -2.0500000000000001e165 < x < 7.40000000000000021e158

   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

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

   4. Applied rewrites 78.9%

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

Alternative 5: 77.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(i, y, x \cdot \log y\right) + z\right) + t\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(b, \log c, z + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (fma i y (* x (log y))) z) t)))
   (if (<= x -1.55e+165)
     t_1
     (if (<= x 7.4e+158) (fma y i (fma b (log c) (+ z 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(i, y, (x * log(y))) + z) + t;
	double tmp;
	if (x <= -1.55e+165) {
		tmp = t_1;
	} else if (x <= 7.4e+158) {
		tmp = fma(y, i, fma(b, log(c), (z + a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(i, y, Float64(x * log(y))) + z) + t)
	tmp = 0.0
	if (x <= -1.55e+165)
		tmp = t_1;
	elseif (x <= 7.4e+158)
		tmp = fma(y, i, fma(b, log(c), Float64(z + 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[(N[(i * y + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[x, -1.55e+165], t$95$1, If[LessEqual[x, 7.4e+158], N[(y * i + N[(b * N[Log[c], $MachinePrecision] + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5500000000000001e165 or 7.40000000000000021e158 < 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 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -1.5500000000000001e165 < x < 7.40000000000000021e158

   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

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   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 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 93.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 77.1%

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

Alternative 6: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, i, x \cdot \log y\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(b, \log c, z + a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y i (* x (log y)))))
   (if (<= x -3e+186)
     t_1
     (if (<= x 7.4e+158) (fma y i (fma b (log c) (+ z 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, (x * log(y)));
	double tmp;
	if (x <= -3e+186) {
		tmp = t_1;
	} else if (x <= 7.4e+158) {
		tmp = fma(y, i, fma(b, log(c), (z + a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, i, Float64(x * log(y)))
	tmp = 0.0
	if (x <= -3e+186)
		tmp = t_1;
	elseif (x <= 7.4e+158)
		tmp = fma(y, i, fma(b, log(c), Float64(z + 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+186], t$95$1, If[LessEqual[x, 7.4e+158], N[(y * i + N[(b * N[Log[c], $MachinePrecision] + N[(z + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999982e186 or 7.40000000000000021e158 < x

   1. Initial program 99.3%

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

   2. 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 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 66.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 66.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 66.4%

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

   10. Taylor expanded in x around inf

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

       1. associate-+l+ N/A

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

       2. *-commutative N/A

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

       3. associate-+l+ N/A

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

       4. +-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-log.f64 69.4

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

   12. Applied rewrites 69.4%

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

   if -2.99999999999999982e186 < x < 7.40000000000000021e158

   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

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 93.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 93.5%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 76.7%

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

Alternative 7: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(z + \log c \cdot b\right) + y \cdot i\\ \mathbf{elif}\;t\_1 \leq 200:\\ \;\;\;\;\left(z + -0.5 \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, \mathsf{fma}\left(b, \log c, t + a\right)\right)\\ \end{array} \end{array} \]

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 -5e+114)
     (+ (+ z (* (log c) b)) (* y i))
     (if (<= t_1 200.0)
       (+ (+ z (* -0.5 (log c))) (* y i))
       (fma y i (fma b (log c) (+ t 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 <= -5e+114) {
		tmp = (z + (log(c) * b)) + (y * i);
	} else if (t_1 <= 200.0) {
		tmp = (z + (-0.5 * log(c))) + (y * i);
	} else {
		tmp = fma(y, i, fma(b, log(c), (t + 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 <= -5e+114)
		tmp = Float64(Float64(z + Float64(log(c) * b)) + Float64(y * i));
	elseif (t_1 <= 200.0)
		tmp = Float64(Float64(z + Float64(-0.5 * log(c))) + Float64(y * i));
	else
		tmp = fma(y, i, fma(b, log(c), Float64(t + 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, -5e+114], N[(N[(z + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 200.0], N[(N[(z + N[(-0.5 * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(y * i + N[(b * N[Log[c], $MachinePrecision] + N[(t + a), $MachinePrecision]), $MachinePrecision]), $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)) < -5.0000000000000001e114

   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 54.5%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -5.0000000000000001e114 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

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

   2. 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 60.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 48.1%

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

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

   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 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 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 87.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 87.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 87.5%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 69.2%

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

Alternative 8: 54.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log c \cdot b\\ t_2 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(z + t\_1\right) + y \cdot i\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\left(z + -0.5 \cdot \log c\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\_1\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b))
        (t_2
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_2 -5e+114)
     (+ (+ z t_1) (* y i))
     (if (<= t_2 200.0)
       (+ (+ z (* -0.5 (log c))) (* y i))
       (+ (+ a t_1) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * b;
	double t_2 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -5e+114) {
		tmp = (z + t_1) + (y * i);
	} else if (t_2 <= 200.0) {
		tmp = (z + (-0.5 * log(c))) + (y * i);
	} else {
		tmp = (a + t_1) + (y * i);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(c) * b
    t_2 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_2 <= (-5d+114)) then
        tmp = (z + t_1) + (y * i)
    else if (t_2 <= 200.0d0) then
        tmp = (z + ((-0.5d0) * log(c))) + (y * i)
    else
        tmp = (a + t_1) + (y * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = Math.log(c) * b;
	double t_2 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_2 <= -5e+114) {
		tmp = (z + t_1) + (y * i);
	} else if (t_2 <= 200.0) {
		tmp = (z + (-0.5 * Math.log(c))) + (y * i);
	} else {
		tmp = (a + t_1) + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = math.log(c) * b
	t_2 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_2 <= -5e+114:
		tmp = (z + t_1) + (y * i)
	elif t_2 <= 200.0:
		tmp = (z + (-0.5 * math.log(c))) + (y * i)
	else:
		tmp = (a + t_1) + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * b)
	t_2 = 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_2 <= -5e+114)
		tmp = Float64(Float64(z + t_1) + Float64(y * i));
	elseif (t_2 <= 200.0)
		tmp = Float64(Float64(z + Float64(-0.5 * log(c))) + Float64(y * i));
	else
		tmp = Float64(Float64(a + t_1) + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = log(c) * b;
	t_2 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_2 <= -5e+114)
		tmp = (z + t_1) + (y * i);
	elseif (t_2 <= 200.0)
		tmp = (z + (-0.5 * log(c))) + (y * i);
	else
		tmp = (a + t_1) + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = 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$2, -5e+114], N[(N[(z + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[(N[(z + N[(-0.5 * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(a + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $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)) < -5.0000000000000001e114

   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 54.5%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -5.0000000000000001e114 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 200

   1. Initial program 99.9%

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

   2. 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 60.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 48.1%

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

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

   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 inf

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

         \[\leadsto \left(a + \log c \cdot b\right) + y \cdot i \]

      3. lift-*.f64 55.0

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

   7. Applied rewrites 55.0%

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

Alternative 9: 54.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (log c) b)))
   (if (<=
        (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
        -100.0)
     (+ (+ z t_1) (* y i))
     (+ (+ a t_1) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = log(c) * b;
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -100.0) {
		tmp = (z + t_1) + (y * i);
	} else {
		tmp = (a + t_1) + (y * i);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(c) * b
    if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)) <= (-100.0d0)) then
        tmp = (z + t_1) + (y * i)
    else
        tmp = (a + t_1) + (y * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = Math.log(c) * b;
	double tmp;
	if (((((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i)) <= -100.0) {
		tmp = (z + t_1) + (y * i);
	} else {
		tmp = (a + t_1) + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = math.log(c) * b
	tmp = 0
	if ((((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)) <= -100.0:
		tmp = (z + t_1) + (y * i)
	else:
		tmp = (a + t_1) + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(log(c) * b)
	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 = Float64(Float64(z + t_1) + Float64(y * i));
	else
		tmp = Float64(Float64(a + t_1) + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = log(c) * b;
	tmp = 0.0;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -100.0)
		tmp = (z + t_1) + (y * i);
	else
		tmp = (a + t_1) + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -100.0], N[(N[(z + t$95$1), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(a + t$95$1), $MachinePrecision] + N[(y * i), $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)) < -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

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 53.6

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

   7. Applied rewrites 53.6%

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

   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 99.6%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

         \[\leadsto \left(a + \log c \cdot b\right) + y \cdot i \]

      3. lift-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

Alternative 10: 53.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y + z\right) + t\\ t_2 := \left(\left(t\_1 + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq -300:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(a + \log c \cdot b\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (* x (log y)) z) t))
        (t_2 (+ (+ (+ t_1 a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 -4e+288)
     (fma y i z)
     (if (<= t_2 -300.0) t_1 (+ (+ a (* (log c) b)) (* y i))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((x * log(y)) + z) + t;
	double t_2 = ((t_1 + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -4e+288) {
		tmp = fma(y, i, z);
	} else if (t_2 <= -300.0) {
		tmp = t_1;
	} else {
		tmp = (a + (log(c) * b)) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(x * log(y)) + z) + t)
	t_2 = Float64(Float64(Float64(t_1 + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -4e+288)
		tmp = fma(y, i, z);
	elseif (t_2 <= -300.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(a + Float64(log(c) * b)) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+288], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, -300.0], t$95$1, N[(N[(a + N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $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)) < -4e288

   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 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 89.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 89.7

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 89.7%

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

   10. Taylor expanded in z around inf

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

       1. associate-+l+ 60.2

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

       2. *-commutative 60.2

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

       3. associate-+l+ 60.2

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

       4. +-commutative 60.2

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

   12. Applied rewrites 60.2%

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

   if -4e288 < (+.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)) < -300

   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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if -300 < (+.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 a around inf

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

         \[\leadsto \left(a + \log c \cdot b\right) + y \cdot i \]

      3. lift-*.f64 54.0

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

   7. Applied rewrites 54.0%

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

Alternative 11: 46.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y + z\right) + t\\ t_2 := \left(\left(t\_1 + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(y, i, z\right)\\ \mathbf{elif}\;t\_2 \leq -300:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, i, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (* x (log y)) z) t))
        (t_2 (+ (+ (+ t_1 a) (* (- b 0.5) (log c))) (* y i))))
   (if (<= t_2 -4e+288) (fma y i z) (if (<= t_2 -300.0) t_1 (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;
	double t_2 = ((t_1 + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_2 <= -4e+288) {
		tmp = fma(y, i, z);
	} else if (t_2 <= -300.0) {
		tmp = t_1;
	} else {
		tmp = fma(y, i, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(x * log(y)) + z) + t)
	t_2 = Float64(Float64(Float64(t_1 + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_2 <= -4e+288)
		tmp = fma(y, i, z);
	elseif (t_2 <= -300.0)
		tmp = t_1;
	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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+288], N[(y * i + z), $MachinePrecision], If[LessEqual[t$95$2, -300.0], t$95$1, 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)) < -4e288

   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 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 89.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 89.7

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 89.7%

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

   10. Taylor expanded in z around inf

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

       1. associate-+l+ 60.2

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

       2. *-commutative 60.2

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

       3. associate-+l+ 60.2

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

       4. +-commutative 60.2

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

   12. Applied rewrites 60.2%

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

   if -4e288 < (+.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)) < -300

   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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if -300 < (+.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 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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 85.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 85.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 85.9%

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

   10. Taylor expanded in a around inf

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

       1. associate-+l+ 39.3

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       2. *-commutative 39.3

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       3. associate-+l+ 39.3

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       4. +-commutative 39.3

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

   12. Applied rewrites 39.3%

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

Alternative 12: 41.4% accurate, 0.8× speedup

Math

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

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))
      -100.0)
   (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)) <= -100.0) {
		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)) <= -100.0)
		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], -100.0], 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)) < -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

      \[\leadsto \left(\left(\left(\color{blue}{z \cdot \left(1 + \frac{x \cdot \log y}{z}\right)} + t\right) + a\right) + \left(b - \frac{1}{2}\right) \cdot \log c\right) + y \cdot i \]
   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 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 88.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 88.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 88.5%

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

   10. Taylor expanded in z around inf

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

       1. associate-+l+ 38.9

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

       2. *-commutative 38.9

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

       3. associate-+l+ 38.9

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

       4. +-commutative 38.9

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

   12. Applied rewrites 38.9%

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

   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 99.6%

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

   2. Taylor expanded in 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 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in a around inf

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

       1. associate-+l+ 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       2. *-commutative 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       3. associate-+l+ 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       4. +-commutative 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

   12. Applied rewrites 39.7%

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

Alternative 13: 41.4% accurate, 0.4× speedup

Math

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

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))
     (fma y i t)
     (if (<= t_1 -100.0) (+ z t) (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 = fma(y, i, t);
	} else if (t_1 <= -100.0) {
		tmp = z + t;
	} 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 = fma(y, i, t);
	elseif (t_1 <= -100.0)
		tmp = Float64(z + t);
	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[(y * i + t), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(z + t), $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 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 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 93.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 93.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 93.5%

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

   10. Taylor expanded in t around inf

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

       1. associate-+l+ 95.8

          \[\leadsto \mathsf{fma}\left(y, i, t\right) \]

       2. *-commutative 95.8

          \[\leadsto \mathsf{fma}\left(y, i, t\right) \]

       3. associate-+l+ 95.8

          \[\leadsto \mathsf{fma}\left(y, i, t\right) \]

       4. +-commutative 95.8

          \[\leadsto \mathsf{fma}\left(y, i, t\right) \]

   12. Applied rewrites 95.8%

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

   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 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto z + t \]
   6. Step-by-step derivation

   7. Applied rewrites 35.7%

      \[\leadsto z + t \]

   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 99.6%

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

   2. Taylor expanded in 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 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in a around inf

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

       1. associate-+l+ 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       2. *-commutative 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       3. associate-+l+ 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       4. +-commutative 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

   12. Applied rewrites 39.7%

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

Alternative 14: 39.3% accurate, 0.4× speedup

Math

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

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 -100.0) (+ z t) (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 <= -100.0) {
		tmp = z + t;
	} 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 <= -100.0)
		tmp = Float64(z + t);
	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, -100.0], N[(z + t), $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 95.8

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

   4. Applied rewrites 95.8%

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

   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 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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto z + t \]
   6. Step-by-step derivation

   7. Applied rewrites 35.7%

      \[\leadsto z + t \]

   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 99.6%

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

   2. Taylor expanded in 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 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 86.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 86.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower-fma.f64 N/A

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in a around inf

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

       1. associate-+l+ 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       2. *-commutative 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       3. associate-+l+ 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

       4. +-commutative 39.7

          \[\leadsto \mathsf{fma}\left(y, i, a\right) \]

   12. Applied rewrites 39.7%

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

Alternative 15: 36.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -50:\\ \;\;\;\;z + t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

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 -50.0) (+ z t) (if (<= t_1 5e+306) 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 <= -50.0) {
		tmp = z + t;
	} else if (t_1 <= 5e+306) {
		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 <= -50.0) {
		tmp = z + t;
	} else if (t_1 <= 5e+306) {
		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 <= -50.0:
		tmp = z + t
	elif t_1 <= 5e+306:
		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 <= -50.0)
		tmp = Float64(z + t);
	elseif (t_1 <= 5e+306)
		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 <= -50.0)
		tmp = z + t;
	elseif (t_1 <= 5e+306)
		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, -50.0], N[(z + t), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], 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 4.99999999999999993e306 < (+.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 99.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 91.4

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

   4. Applied rewrites 91.4%

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

   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)) < -50

   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. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto z + t \]
   6. Step-by-step derivation

   7. Applied rewrites 35.6%

      \[\leadsto z + t \]

   if -50 < (+.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)) < 4.99999999999999993e306

   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 18.7%

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

Alternative 16: 16.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -50:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

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))
      -50.0)
   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)) <= -50.0) {
		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)) <= (-50.0d0)) 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)) <= -50.0) {
		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)) <= -50.0:
		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)) <= -50.0)
		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)) <= -50.0)
		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], -50.0], 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)) < -50

   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 17.3%

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

   if -50 < (+.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 a around inf

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

   4. Applied rewrites 16.3%

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

Alternative 17: 16.7% accurate, 37.6× 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 99.7%

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

2. Taylor expanded in a around inf

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

4. Applied rewrites 16.7%

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

Reproduce

herbie shell --seed 2025114 
(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)))