Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 56.3% → 86.3%

Time: 10.0s

Alternatives: 15

Speedup: 2.3×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + 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 * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + 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 * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Initial Program: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + 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 * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + 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 * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), \frac{y}{t\_1}, \frac{27464.7644705}{t\_1}\right), \frac{230661.510616}{t\_1}\right), y, \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<=
        (/
         (+
          (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
          t)
         (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
        INFINITY)
     (fma
      (fma
       y
       (fma (fma x y z) (/ y t_1) (/ 27464.7644705 t_1))
       (/ 230661.510616 t_1))
      y
      (/ t t_1))
     (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma(fma(y, fma(fma(x, y, z), (y / t_1), (27464.7644705 / t_1)), (230661.510616 / t_1)), y, (t / t_1));
	} else {
		tmp = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = fma(fma(y, fma(fma(x, y, z), Float64(y / t_1), Float64(27464.7644705 / t_1)), Float64(230661.510616 / t_1)), y, Float64(t / t_1));
	else
		tmp = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y * N[(N[(x * y + z), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision] + N[(27464.7644705 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 / t$95$1), $MachinePrecision]), $MachinePrecision] * y + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      3. div-add N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right) \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} + \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

   5. Applied rewrites 57.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{230661.510616}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, \frac{54929528941}{2000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}}, \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \frac{\color{blue}{\mathsf{fma}\left(x, y, z\right) \cdot y + \frac{54929528941}{2000000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      3. div-add N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{\frac{\mathsf{fma}\left(x, y, z\right) \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} + \frac{\frac{54929528941}{2000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}}, \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(x, y, z\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} + \frac{\frac{54929528941}{2000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{\frac{54929528941}{2000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)}, \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), \color{blue}{\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}}, \frac{\frac{54929528941}{2000000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), \frac{\frac{28832688827}{125000}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      7. lower-/.f64 59.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \color{blue}{\frac{27464.7644705}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}}\right), \frac{230661.510616}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

   7. Applied rewrites 59.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{27464.7644705}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)}, \frac{230661.510616}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right), y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_1}, y, \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<=
        (/
         (+
          (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
          t)
         (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
        INFINITY)
     (fma
      (/ (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) t_1)
      y
      (/ t t_1))
     (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma((fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), y, (t / t_1));
	} else {
		tmp = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = fma(Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), y, Float64(t / t_1));
	else
		tmp = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] * y + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \mathbf{if}\;y \leq -2.75 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))
   (if (<= y -2.75e+20)
     t_1
     (if (<= y 9.5e+39)
       (/
        (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
        (fma (fma (fma a y b) y 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 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.75e+20) {
		tmp = t_1;
	} else if (y <= 9.5e+39) {
		tmp = fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(a, y, b), y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)))
	tmp = 0.0
	if (y <= -2.75e+20)
		tmp = t_1;
	elseif (y <= 9.5e+39)
		tmp = Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(a, y, b), y, c), 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[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.75e+20], t$95$1, If[LessEqual[y, 9.5e+39], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(a * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.75e20 or 9.50000000000000011e39 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   if -2.75e20 < y < 9.50000000000000011e39

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. lower-fma.f64 56.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. lower-fma.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705, y, 230661.510616\right)}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y + z\right) \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      9. lower-fma.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot y + z, y, 27464.7644705\right)}, y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y + z}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y} + z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot x} + z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      13. lower-fma.f64 56.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, x, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

   3. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a}, y, b\right), y, c\right), y, i\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 54.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a}, y, b\right), y, c\right), y, i\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_1}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<=
        (/
         (+
          (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
          t)
         (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
        INFINITY)
     (fma
      (/ y t_1)
      (fma (fma (fma y x z) y 27464.7644705) y 230661.510616)
      (/ t t_1))
     (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma((y / t_1), fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), (t / t_1));
	} else {
		tmp = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = fma(Float64(y / t_1), fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), Float64(t / t_1));
	else
		tmp = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(y / t$95$1), $MachinePrecision] * N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot \frac{y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}, \left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}, \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\right)} \]

   3. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
      INFINITY)
   (/
    (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
    (fma (fma (fma (+ a y) y b) y c) y i))
   (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. lower-fma.f64 56.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. lower-fma.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705, y, 230661.510616\right)}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y + z\right) \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      9. lower-fma.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot y + z, y, 27464.7644705\right)}, y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y + z}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y} + z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot x} + z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      13. lower-fma.f64 56.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, x, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

   3. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \mathbf{if}\;y \leq -2.75 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))
   (if (<= y -2.75e+20)
     t_1
     (if (<= y 7.8e+39)
       (/
        (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
        (fma (fma b y 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 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.75e+20) {
		tmp = t_1;
	} else if (y <= 7.8e+39) {
		tmp = fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(b, y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)))
	tmp = 0.0
	if (y <= -2.75e+20)
		tmp = t_1;
	elseif (y <= 7.8e+39)
		tmp = Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(b, y, c), 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[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.75e+20], t$95$1, If[LessEqual[y, 7.8e+39], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(b * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.75e20 or 7.8000000000000002e39 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   if -2.75e20 < y < 7.8000000000000002e39

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. lower-fma.f64 56.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y} + \frac{28832688827}{125000}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. lower-fma.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705, y, 230661.510616\right)}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}}, y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot y + z\right) \cdot y} + \frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      9. lower-fma.f64 56.3

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot y + z, y, 27464.7644705\right)}, y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y + z}, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot y} + z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y \cdot x} + z, y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      13. lower-fma.f64 56.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, x, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

   3. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b}, y, c\right), y, i\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{b}, y, c\right), y, i\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))
   (if (<= y -2.5e+20)
     t_1
     (if (<= y 2.9e+39)
       (/ (fma 230661.510616 y t) (fma (fma (fma (+ a y) y b) y 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 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.5e+20) {
		tmp = t_1;
	} else if (y <= 2.9e+39) {
		tmp = fma(230661.510616, y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)))
	tmp = 0.0
	if (y <= -2.5e+20)
		tmp = t_1;
	elseif (y <= 2.9e+39)
		tmp = Float64(fma(230661.510616, y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), 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[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+20], t$95$1, If[LessEqual[y, 2.9e+39], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e20 or 2.90000000000000029e39 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   if -2.5e20 < y < 2.90000000000000029e39

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.1%

      \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. lower-fma.f64 48.1

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y} + i} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\color{blue}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} \cdot y + i} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\left(\color{blue}{\left(\left(y + a\right) \cdot y + b\right) \cdot y} + c\right) \cdot y + i} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\color{blue}{\mathsf{fma}\left(\left(y + a\right) \cdot y + b, y, c\right)} \cdot y + i} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\mathsf{fma}\left(\color{blue}{\left(y + a\right) \cdot y + b}, y, c\right) \cdot y + i} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\mathsf{fma}\left(\color{blue}{\left(y + a\right) \cdot y} + b, y, c\right) \cdot y + i} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y + a, y, b\right)}, y, c\right) \cdot y + i} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{y + a}, y, b\right), y, c\right) \cdot y + i} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right) \cdot y + i} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a + y}, y, b\right), y, c\right) \cdot y + i} \]

   6. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right) \cdot y + c, y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))
   (if (<= y -2.5e+20)
     t_1
     (if (<= y 5.6e+39)
       (/ (fma 230661.510616 y t) (fma (+ (* (fma y y b) y) 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 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.5e+20) {
		tmp = t_1;
	} else if (y <= 5.6e+39) {
		tmp = fma(230661.510616, y, t) / fma(((fma(y, y, b) * y) + c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)))
	tmp = 0.0
	if (y <= -2.5e+20)
		tmp = t_1;
	elseif (y <= 5.6e+39)
		tmp = Float64(fma(230661.510616, y, t) / fma(Float64(Float64(fma(y, y, b) * y) + c), 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[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+20], t$95$1, If[LessEqual[y, 5.6e+39], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(y * y + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e20 or 5.60000000000000003e39 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   if -2.5e20 < y < 5.60000000000000003e39

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.1%

      \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\left(\color{blue}{y \cdot \left(b + {y}^{2}\right)} + c\right) \cdot y + i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\left(y \cdot \color{blue}{\left(b + {y}^{2}\right)} + c\right) \cdot y + i} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\left(y \cdot \left(b + \color{blue}{{y}^{2}}\right) + c\right) \cdot y + i} \]

      3. lower-pow.f64 46.3

         \[\leadsto \frac{230661.510616 \cdot y + t}{\left(y \cdot \left(b + {y}^{\color{blue}{2}}\right) + c\right) \cdot y + i} \]

   7. Applied rewrites 46.3%

      \[\leadsto \frac{230661.510616 \cdot y + t}{\left(\color{blue}{y \cdot \left(b + {y}^{2}\right)} + c\right) \cdot y + i} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y + t}}{\left(y \cdot \left(b + {y}^{2}\right) + c\right) \cdot y + i} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y} + t}{\left(y \cdot \left(b + {y}^{2}\right) + c\right) \cdot y + i} \]

      3. lower-fma.f64 46.3

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(y \cdot \left(b + {y}^{2}\right) + c\right) \cdot y + i} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\color{blue}{\left(y \cdot \left(b + {y}^{2}\right) + c\right) \cdot y + i}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{\color{blue}{\left(y \cdot \left(b + {y}^{2}\right) + c\right) \cdot y} + i} \]

      6. lower-fma.f64 46.3

         \[\leadsto \frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\color{blue}{\mathsf{fma}\left(y \cdot \left(b + {y}^{2}\right) + c, y, i\right)}} \]

   9. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, y, b\right) \cdot y + c, y, i\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+38}:\\ \;\;\;\;\frac{230661.510616 \cdot y + t}{\left(b \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))
   (if (<= y -2.5e+20)
     t_1
     (if (<= y 1.55e+38)
       (/ (+ (* 230661.510616 y) t) (+ (* (+ (* b y) 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 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.5e+20) {
		tmp = t_1;
	} else if (y <= 1.55e+38) {
		tmp = ((230661.510616 * y) + t) / ((((b * y) + c) * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + ((-1.0d0) * ((z * ((a / y) - 1.0d0)) / y))
    if (y <= (-2.5d+20)) then
        tmp = t_1
    else if (y <= 1.55d+38) then
        tmp = ((230661.510616d0 * y) + t) / ((((b * y) + c) * y) + i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.5e+20) {
		tmp = t_1;
	} else if (y <= 1.55e+38) {
		tmp = ((230661.510616 * y) + t) / ((((b * y) + c) * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y))
	tmp = 0
	if y <= -2.5e+20:
		tmp = t_1
	elif y <= 1.55e+38:
		tmp = ((230661.510616 * y) + t) / ((((b * y) + c) * y) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)))
	tmp = 0.0
	if (y <= -2.5e+20)
		tmp = t_1;
	elseif (y <= 1.55e+38)
		tmp = Float64(Float64(Float64(230661.510616 * y) + t) / Float64(Float64(Float64(Float64(b * y) + c) * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	tmp = 0.0;
	if (y <= -2.5e+20)
		tmp = t_1;
	elseif (y <= 1.55e+38)
		tmp = ((230661.510616 * y) + t) / ((((b * y) + c) * y) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+20], t$95$1, If[LessEqual[y, 1.55e+38], N[(N[(N[(230661.510616 * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(b * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5e20 or 1.55000000000000009e38 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   if -2.5e20 < y < 1.55000000000000009e38

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.1%

      \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\left(\color{blue}{b \cdot y} + c\right) \cdot y + i} \]
   6. Step-by-step derivation

      1. lower-*.f64 45.9

         \[\leadsto \frac{230661.510616 \cdot y + t}{\left(b \cdot \color{blue}{y} + c\right) \cdot y + i} \]

   7. Applied rewrites 45.9%

      \[\leadsto \frac{230661.510616 \cdot y + t}{\left(\color{blue}{b \cdot y} + c\right) \cdot y + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+38}:\\ \;\;\;\;\frac{230661.510616 \cdot y + t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (* -1.0 (/ (* z (- (/ a y) 1.0)) y)))))
   (if (<= y -2.4e+20)
     t_1
     (if (<= y 1.55e+38) (/ (+ (* 230661.510616 y) t) (+ (* 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 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.4e+20) {
		tmp = t_1;
	} else if (y <= 1.55e+38) {
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + ((-1.0d0) * ((z * ((a / y) - 1.0d0)) / y))
    if (y <= (-2.4d+20)) then
        tmp = t_1
    else if (y <= 1.55d+38) then
        tmp = ((230661.510616d0 * y) + t) / ((c * y) + i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	double tmp;
	if (y <= -2.4e+20) {
		tmp = t_1;
	} else if (y <= 1.55e+38) {
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y))
	tmp = 0
	if y <= -2.4e+20:
		tmp = t_1
	elif y <= 1.55e+38:
		tmp = ((230661.510616 * y) + t) / ((c * y) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(-1.0 * Float64(Float64(z * Float64(Float64(a / y) - 1.0)) / y)))
	tmp = 0.0
	if (y <= -2.4e+20)
		tmp = t_1;
	elseif (y <= 1.55e+38)
		tmp = Float64(Float64(Float64(230661.510616 * y) + t) / Float64(Float64(c * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + (-1.0 * ((z * ((a / y) - 1.0)) / y));
	tmp = 0.0;
	if (y <= -2.4e+20)
		tmp = t_1;
	elseif (y <= 1.55e+38)
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(-1.0 * N[(N[(z * N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+20], t$95$1, If[LessEqual[y, 1.55e+38], N[(N[(N[(230661.510616 * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e20 or 1.55000000000000009e38 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

      3. lower-/.f64 33.4

         \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   9. Applied rewrites 33.4%

      \[\leadsto x + -1 \cdot \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]

   if -2.4e20 < y < 1.55000000000000009e38

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.1%

      \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\color{blue}{c \cdot y} + i} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.2

         \[\leadsto \frac{230661.510616 \cdot y + t}{c \cdot \color{blue}{y} + i} \]

   7. Applied rewrites 42.2%

      \[\leadsto \frac{230661.510616 \cdot y + t}{\color{blue}{c \cdot y} + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{230661.510616 \cdot y + t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -6.8e+19)
     t_1
     (if (<= y 5.4e-10) (/ (+ (* 230661.510616 y) t) (+ (* 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 = x + ((z - (a * x)) / y);
	double tmp;
	if (y <= -6.8e+19) {
		tmp = t_1;
	} else if (y <= 5.4e-10) {
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + ((z - (a * x)) / y)
    if (y <= (-6.8d+19)) then
        tmp = t_1
    else if (y <= 5.4d-10) then
        tmp = ((230661.510616d0 * y) + t) / ((c * y) + i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (a * x)) / y);
	double tmp;
	if (y <= -6.8e+19) {
		tmp = t_1;
	} else if (y <= 5.4e-10) {
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + ((z - (a * x)) / y)
	tmp = 0
	if y <= -6.8e+19:
		tmp = t_1
	elif y <= 5.4e-10:
		tmp = ((230661.510616 * y) + t) / ((c * y) + i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -6.8e+19)
		tmp = t_1;
	elseif (y <= 5.4e-10)
		tmp = Float64(Float64(Float64(230661.510616 * y) + t) / Float64(Float64(c * y) + i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + ((z - (a * x)) / y);
	tmp = 0.0;
	if (y <= -6.8e+19)
		tmp = t_1;
	elseif (y <= 5.4e-10)
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e+19], t$95$1, If[LessEqual[y, 5.4e-10], N[(N[(N[(230661.510616 * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.8e19 or 5.4e-10 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x + \frac{z - a \cdot x}{\color{blue}{y}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

      3. lower-*.f64 31.3

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

   9. Applied rewrites 31.3%

      \[\leadsto x + \frac{z - a \cdot x}{\color{blue}{y}} \]

   if -6.8e19 < y < 5.4e-10

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.1%

      \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\color{blue}{c \cdot y} + i} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.2

         \[\leadsto \frac{230661.510616 \cdot y + t}{c \cdot \color{blue}{y} + i} \]

   7. Applied rewrites 42.2%

      \[\leadsto \frac{230661.510616 \cdot y + t}{\color{blue}{c \cdot y} + i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{230661.510616 \cdot y + t}{c \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -6.8e+19)
   (+ x (/ (- z (* a x)) y))
   (if (<= y 5.4e-10)
     (/ (+ (* 230661.510616 y) t) (+ (* c y) i))
     (- (+ x (/ z y)) (/ (* a x) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -6.8e+19) {
		tmp = x + ((z - (a * x)) / y);
	} else if (y <= 5.4e-10) {
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	} else {
		tmp = (x + (z / y)) - ((a * x) / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-6.8d+19)) then
        tmp = x + ((z - (a * x)) / y)
    else if (y <= 5.4d-10) then
        tmp = ((230661.510616d0 * y) + t) / ((c * y) + i)
    else
        tmp = (x + (z / y)) - ((a * x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -6.8e+19) {
		tmp = x + ((z - (a * x)) / y);
	} else if (y <= 5.4e-10) {
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	} else {
		tmp = (x + (z / y)) - ((a * x) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -6.8e+19:
		tmp = x + ((z - (a * x)) / y)
	elif y <= 5.4e-10:
		tmp = ((230661.510616 * y) + t) / ((c * y) + i)
	else:
		tmp = (x + (z / y)) - ((a * x) / y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -6.8e+19)
		tmp = Float64(x + Float64(Float64(z - Float64(a * x)) / y));
	elseif (y <= 5.4e-10)
		tmp = Float64(Float64(Float64(230661.510616 * y) + t) / Float64(Float64(c * y) + i));
	else
		tmp = Float64(Float64(x + Float64(z / y)) - Float64(Float64(a * x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -6.8e+19)
		tmp = x + ((z - (a * x)) / y);
	elseif (y <= 5.4e-10)
		tmp = ((230661.510616 * y) + t) / ((c * y) + i);
	else
		tmp = (x + (z / y)) - ((a * x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -6.8e+19], N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-10], N[(N[(N[(230661.510616 * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(c * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(N[(a * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8e19

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x + \frac{z - a \cdot x}{\color{blue}{y}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

      3. lower-*.f64 31.3

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

   9. Applied rewrites 31.3%

      \[\leadsto x + \frac{z - a \cdot x}{\color{blue}{y}} \]

   if -6.8e19 < y < 5.4e-10

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000}} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.1%

      \[\leadsto \frac{\color{blue}{230661.510616} \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{28832688827}{125000} \cdot y + t}{\color{blue}{c \cdot y} + i} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.2

         \[\leadsto \frac{230661.510616 \cdot y + t}{c \cdot \color{blue}{y} + i} \]

   7. Applied rewrites 42.2%

      \[\leadsto \frac{230661.510616 \cdot y + t}{\color{blue}{c \cdot y} + i} \]

   if 5.4e-10 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(x + \frac{z}{y}\right) - \color{blue}{\frac{a \cdot x}{y}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(x + \frac{z}{y}\right) - \frac{\color{blue}{a \cdot x}}{y} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(x + \frac{z}{y}\right) - \frac{a \cdot \color{blue}{x}}{y} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(x + \frac{z}{y}\right) - \frac{a \cdot x}{\color{blue}{y}} \]

      5. lower-*.f64 31.2

         \[\leadsto \left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y} \]

   4. Applied rewrites 31.2%

      \[\leadsto \color{blue}{\left(x + \frac{z}{y}\right) - \frac{a \cdot x}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{t + 230661.510616 \cdot y}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -2.25e-34)
     t_1
     (if (<= y 3e-14) (/ (+ t (* 230661.510616 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 = x + ((z - (a * x)) / y);
	double tmp;
	if (y <= -2.25e-34) {
		tmp = t_1;
	} else if (y <= 3e-14) {
		tmp = (t + (230661.510616 * y)) / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + ((z - (a * x)) / y)
    if (y <= (-2.25d-34)) then
        tmp = t_1
    else if (y <= 3d-14) then
        tmp = (t + (230661.510616d0 * y)) / i
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (a * x)) / y);
	double tmp;
	if (y <= -2.25e-34) {
		tmp = t_1;
	} else if (y <= 3e-14) {
		tmp = (t + (230661.510616 * y)) / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + ((z - (a * x)) / y)
	tmp = 0
	if y <= -2.25e-34:
		tmp = t_1
	elif y <= 3e-14:
		tmp = (t + (230661.510616 * y)) / i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -2.25e-34)
		tmp = t_1;
	elseif (y <= 3e-14)
		tmp = Float64(Float64(t + Float64(230661.510616 * y)) / i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + ((z - (a * x)) / y);
	tmp = 0.0;
	if (y <= -2.25e-34)
		tmp = t_1;
	elseif (y <= 3e-14)
		tmp = (t + (230661.510616 * y)) / i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.25e-34], t$95$1, If[LessEqual[y, 3e-14], N[(N[(t + N[(230661.510616 * y), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.25000000000000021e-34 or 2.9999999999999998e-14 < y

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} + \frac{t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   3. Applied rewrites 57.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, y, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)} \]

   4. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + \color{blue}{-1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + -1 \cdot \color{blue}{\frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{y}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + -1 \cdot \frac{-1 \cdot \left(z - a \cdot x\right) + -1 \cdot \frac{\frac{54929528941}{2000000} + \left(-1 \cdot \left(a \cdot \left(z - a \cdot x\right)\right) + -1 \cdot \left(b \cdot x\right)\right)}{y}}{\color{blue}{y}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\mathsf{fma}\left(-1, z - a \cdot x, -1 \cdot \frac{27464.7644705 + \mathsf{fma}\left(-1, a \cdot \left(z - a \cdot x\right), -1 \cdot \left(b \cdot x\right)\right)}{y}\right)}{y}} \]

   7. Taylor expanded in y around inf

      \[\leadsto x + \frac{z - a \cdot x}{\color{blue}{y}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

      2. lower--.f64 N/A

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

      3. lower-*.f64 31.3

         \[\leadsto x + \frac{z - a \cdot x}{y} \]

   9. Applied rewrites 31.3%

      \[\leadsto x + \frac{z - a \cdot x}{\color{blue}{y}} \]

   if -2.25000000000000021e-34 < y < 2.9999999999999998e-14

   1. Initial program 56.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right) + \frac{t}{i}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}}, \frac{t}{i}\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \color{blue}{\frac{c \cdot t}{{i}^{2}}}, \frac{t}{i}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{\color{blue}{c \cdot t}}{{i}^{2}}, \frac{t}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot \color{blue}{t}}{{i}^{2}}, \frac{t}{i}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{\color{blue}{{i}^{2}}}, \frac{t}{i}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{\color{blue}{i}}^{2}}, \frac{t}{i}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{\color{blue}{2}}}, \frac{t}{i}\right) \]

      8. lower-/.f64 25.3

         \[\leadsto \mathsf{fma}\left(y, 230661.510616 \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}, \frac{t}{i}\right) \]

   4. Applied rewrites 25.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 230661.510616 \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}, \frac{t}{i}\right)} \]

   5. Taylor expanded in i around inf

      \[\leadsto \frac{t + \frac{28832688827}{125000} \cdot y}{\color{blue}{i}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + \frac{28832688827}{125000} \cdot y}{i} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{t + \frac{28832688827}{125000} \cdot y}{i} \]

      3. lower-*.f64 31.8

         \[\leadsto \frac{t + 230661.510616 \cdot y}{i} \]

   7. Applied rewrites 31.8%

      \[\leadsto \frac{t + 230661.510616 \cdot y}{\color{blue}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{t + 230661.510616 \cdot y}{i} \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (/ (+ t (* 230661.510616 y)) i))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (t + (230661.510616 * 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 = (t + (230661.510616d0 * 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 (t + (230661.510616 * y)) / i;
}

Python

def code(x, y, z, t, a, b, c, i):
	return (t + (230661.510616 * y)) / i

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(t + Float64(230661.510616 * y)) / i)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (t + (230661.510616 * y)) / i;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(t + N[(230661.510616 * y), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right) + \frac{t}{i}} \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}}, \frac{t}{i}\right) \]

   2. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \color{blue}{\frac{c \cdot t}{{i}^{2}}}, \frac{t}{i}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{\color{blue}{c \cdot t}}{{i}^{2}}, \frac{t}{i}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot \color{blue}{t}}{{i}^{2}}, \frac{t}{i}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{\color{blue}{{i}^{2}}}, \frac{t}{i}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{\color{blue}{i}}^{2}}, \frac{t}{i}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, \frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{\color{blue}{2}}}, \frac{t}{i}\right) \]

   8. lower-/.f64 25.3

      \[\leadsto \mathsf{fma}\left(y, 230661.510616 \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}, \frac{t}{i}\right) \]

4. Applied rewrites 25.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 230661.510616 \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}, \frac{t}{i}\right)} \]

5. Taylor expanded in i around inf

   \[\leadsto \frac{t + \frac{28832688827}{125000} \cdot y}{\color{blue}{i}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t + \frac{28832688827}{125000} \cdot y}{i} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{t + \frac{28832688827}{125000} \cdot y}{i} \]

   3. lower-*.f64 31.8

      \[\leadsto \frac{t + 230661.510616 \cdot y}{i} \]

7. Applied rewrites 31.8%

   \[\leadsto \frac{t + 230661.510616 \cdot y}{\color{blue}{i}} \]
8. Add Preprocessing

Alternative 15: 28.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{t}{i} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

2. Taylor expanded in y around 0

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

   1. lower-/.f64 28.7

      \[\leadsto \frac{t}{\color{blue}{i}} \]

4. Applied rewrites 28.7%

   \[\leadsto \color{blue}{\frac{t}{i}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))