Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 55.7% → 83.1%

Time: 9.7s

Alternatives: 24

Speedup: 1.7×

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: 55.7% 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: 83.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ t_2 := c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\\ t_3 := {t\_2}^{2}\\ t_4 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ t_5 := y \cdot t\_2\\ t_6 := z + x \cdot y\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_1}, \frac{t}{t\_1}\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-1, i \cdot \mathsf{fma}\left(27464.7644705, \frac{1}{y \cdot t\_3}, \mathsf{fma}\left(230661.510616, \frac{1}{{t\_5}^{2}}, \frac{t\_6}{t\_3}\right)\right), \mathsf{fma}\left(230661.510616, \frac{1}{t\_5}, \frac{27464.7644705 + y \cdot t\_6}{t\_2}\right)\right), \frac{t}{\mathsf{fma}\left(c + \left(y \cdot y\right) \cdot \left(a + y\right), y, i\right)}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(\frac{27464.7644705}{a \cdot \left(x \cdot y\right)} + \left(\frac{230661.510616}{a \cdot \left(x \cdot \left(y \cdot y\right)\right)} + \left(\frac{t}{a \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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))
        (t_2 (+ c (* y (+ b (* y (+ a y))))))
        (t_3 (pow t_2 2.0))
        (t_4 (+ (- (/ (- (- z) (* (- a) x)) y)) x))
        (t_5 (* y t_2))
        (t_6 (+ z (* x y))))
   (if (<= y -3.8e+47)
     t_4
     (if (<= y 2.4e+32)
       (fma
        y
        (/ (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) t_1)
        (/ t t_1))
       (if (<= y 4e+71)
         (fma
          y
          (fma
           -1.0
           (*
            i
            (fma
             27464.7644705
             (/ 1.0 (* y t_3))
             (fma 230661.510616 (/ 1.0 (pow t_5 2.0)) (/ t_6 t_3))))
           (fma 230661.510616 (/ 1.0 t_5) (/ (+ 27464.7644705 (* y t_6)) t_2)))
          (/ t (fma (+ c (* (* y y) (+ a y))) y i)))
         (if (<= y 1.4e+105)
           (*
            x
            (+
             (/ 27464.7644705 (* a (* x y)))
             (+
              (/ 230661.510616 (* a (* x (* y y))))
              (+ (/ t (* a (* x (* (* y y) y)))) (+ (/ y a) (/ z (* a x)))))))
           t_4))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double t_2 = c + (y * (b + (y * (a + y))));
	double t_3 = pow(t_2, 2.0);
	double t_4 = -((-z - (-a * x)) / y) + x;
	double t_5 = y * t_2;
	double t_6 = z + (x * y);
	double tmp;
	if (y <= -3.8e+47) {
		tmp = t_4;
	} else if (y <= 2.4e+32) {
		tmp = fma(y, (fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), (t / t_1));
	} else if (y <= 4e+71) {
		tmp = fma(y, fma(-1.0, (i * fma(27464.7644705, (1.0 / (y * t_3)), fma(230661.510616, (1.0 / pow(t_5, 2.0)), (t_6 / t_3)))), fma(230661.510616, (1.0 / t_5), ((27464.7644705 + (y * t_6)) / t_2))), (t / fma((c + ((y * y) * (a + y))), y, i)));
	} else if (y <= 1.4e+105) {
		tmp = x * ((27464.7644705 / (a * (x * y))) + ((230661.510616 / (a * (x * (y * y)))) + ((t / (a * (x * ((y * y) * y)))) + ((y / a) + (z / (a * x))))));
	} else {
		tmp = t_4;
	}
	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)
	t_2 = Float64(c + Float64(y * Float64(b + Float64(y * Float64(a + y)))))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	t_5 = Float64(y * t_2)
	t_6 = Float64(z + Float64(x * y))
	tmp = 0.0
	if (y <= -3.8e+47)
		tmp = t_4;
	elseif (y <= 2.4e+32)
		tmp = fma(y, Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), Float64(t / t_1));
	elseif (y <= 4e+71)
		tmp = fma(y, fma(-1.0, Float64(i * fma(27464.7644705, Float64(1.0 / Float64(y * t_3)), fma(230661.510616, Float64(1.0 / (t_5 ^ 2.0)), Float64(t_6 / t_3)))), fma(230661.510616, Float64(1.0 / t_5), Float64(Float64(27464.7644705 + Float64(y * t_6)) / t_2))), Float64(t / fma(Float64(c + Float64(Float64(y * y) * Float64(a + y))), y, i)));
	elseif (y <= 1.4e+105)
		tmp = Float64(x * Float64(Float64(27464.7644705 / Float64(a * Float64(x * y))) + Float64(Float64(230661.510616 / Float64(a * Float64(x * Float64(y * y)))) + Float64(Float64(t / Float64(a * Float64(x * Float64(Float64(y * y) * y)))) + Float64(Float64(y / a) + Float64(z / Float64(a * x)))))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, Block[{t$95$2 = N[(c + N[(y * N[(b + N[(y * N[(a + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, Block[{t$95$5 = N[(y * t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e+47], t$95$4, If[LessEqual[y, 2.4e+32], N[(y * N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+71], N[(y * N[(-1.0 * N[(i * N[(27464.7644705 * N[(1.0 / N[(y * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 * N[(1.0 / N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$6 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 * N[(1.0 / t$95$5), $MachinePrecision] + N[(N[(27464.7644705 + N[(y * t$95$6), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(c + N[(N[(y * y), $MachinePrecision] * N[(a + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+105], N[(x * N[(N[(27464.7644705 / N[(a * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(230661.510616 / N[(a * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(a * N[(x * N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / a), $MachinePrecision] + N[(z / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.8000000000000003e47 or 1.4000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.8000000000000003e47 < y < 2.39999999999999991e32

   1. Initial program 55.7%

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

   3. Applied rewrites 56.4%

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

   if 2.39999999999999991e32 < y < 4.0000000000000002e71

   1. Initial program 55.7%

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

   3. Applied rewrites 56.4%

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

   4. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(\color{blue}{c + {y}^{2} \cdot \left(a + y\right)}, y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(c + \color{blue}{{y}^{2} \cdot \left(a + y\right)}, y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(c + {y}^{2} \cdot \color{blue}{\left(a + y\right)}, y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 52.3

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(c + \left(y \cdot y\right) \cdot \left(a + y\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\color{blue}{c + {y}^{2} \cdot \left(a + y\right)}, y, i\right)}\right) \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(c + \left(y \cdot y\right) \cdot \left(a + y\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(c + \color{blue}{{y}^{2} \cdot \left(a + y\right)}, y, i\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000}\right)}{\mathsf{fma}\left(c + \left(y \cdot y\right) \cdot \left(a + y\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(c + {y}^{2} \cdot \color{blue}{\left(a + y\right)}, y, i\right)}\right) \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 50.6

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

   9. Applied rewrites 50.6%

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

   10. Taylor expanded in i around 0

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1 \cdot \left(i \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{y \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \left(\frac{28832688827}{125000} \cdot \frac{1}{{y}^{2} \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \left(\frac{z}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \frac{x \cdot y}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}\right)\right)\right)\right) + \left(\frac{28832688827}{125000} \cdot \frac{1}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}\right)}, \frac{t}{\mathsf{fma}\left(c + \left(y \cdot y\right) \cdot \left(a + y\right), y, i\right)}\right) \]

   12. Applied rewrites 27.1%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(-1, i \cdot \mathsf{fma}\left(27464.7644705, \frac{1}{y \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}, \mathsf{fma}\left(230661.510616, \frac{1}{{\left(y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)\right)}^{2}}, \frac{z + x \cdot y}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}\right)\right), \mathsf{fma}\left(230661.510616, \frac{1}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{27464.7644705 + y \cdot \left(z + x \cdot y\right)}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}\right)\right)}, \frac{t}{\mathsf{fma}\left(c + \left(y \cdot y\right) \cdot \left(a + y\right), y, i\right)}\right) \]

   if 4.0000000000000002e71 < y < 1.4000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \color{blue}{\left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \color{blue}{\left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\color{blue}{\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)}} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{\color{blue}{a \cdot \left(x \cdot {y}^{3}\right)}} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \color{blue}{\left(x \cdot {y}^{3}\right)}} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \color{blue}{\left(\frac{y}{a} + \frac{z}{a \cdot x}\right)}\right)\right)\right) \]

   7. Applied rewrites 13.5%

      \[\leadsto x \cdot \color{blue}{\left(\frac{27464.7644705}{a \cdot \left(x \cdot y\right)} + \left(\frac{230661.510616}{a \cdot \left(x \cdot \left(y \cdot y\right)\right)} + \left(\frac{t}{a \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ t_2 := c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\\ t_3 := {t\_2}^{2}\\ t_4 := \frac{t}{t\_1}\\ t_5 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ t_6 := y \cdot t\_2\\ t_7 := z + x \cdot y\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_1}, t\_4\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-1, i \cdot \mathsf{fma}\left(27464.7644705, \frac{1}{y \cdot t\_3}, \mathsf{fma}\left(230661.510616, \frac{1}{{t\_6}^{2}}, \frac{t\_7}{t\_3}\right)\right), \mathsf{fma}\left(230661.510616, \frac{1}{t\_6}, \frac{27464.7644705 + y \cdot t\_7}{t\_2}\right)\right), t\_4\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(\frac{27464.7644705}{a \cdot \left(x \cdot y\right)} + \left(\frac{230661.510616}{a \cdot \left(x \cdot \left(y \cdot y\right)\right)} + \left(\frac{t}{a \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

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))
        (t_2 (+ c (* y (+ b (* y (+ a y))))))
        (t_3 (pow t_2 2.0))
        (t_4 (/ t t_1))
        (t_5 (+ (- (/ (- (- z) (* (- a) x)) y)) x))
        (t_6 (* y t_2))
        (t_7 (+ z (* x y))))
   (if (<= y -3.8e+47)
     t_5
     (if (<= y 8.5e+30)
       (fma
        y
        (/ (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) t_1)
        t_4)
       (if (<= y 4e+71)
         (fma
          y
          (fma
           -1.0
           (*
            i
            (fma
             27464.7644705
             (/ 1.0 (* y t_3))
             (fma 230661.510616 (/ 1.0 (pow t_6 2.0)) (/ t_7 t_3))))
           (fma 230661.510616 (/ 1.0 t_6) (/ (+ 27464.7644705 (* y t_7)) t_2)))
          t_4)
         (if (<= y 1.4e+105)
           (*
            x
            (+
             (/ 27464.7644705 (* a (* x y)))
             (+
              (/ 230661.510616 (* a (* x (* y y))))
              (+ (/ t (* a (* x (* (* y y) y)))) (+ (/ y a) (/ z (* a x)))))))
           t_5))))))

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 t_2 = c + (y * (b + (y * (a + y))));
	double t_3 = pow(t_2, 2.0);
	double t_4 = t / t_1;
	double t_5 = -((-z - (-a * x)) / y) + x;
	double t_6 = y * t_2;
	double t_7 = z + (x * y);
	double tmp;
	if (y <= -3.8e+47) {
		tmp = t_5;
	} else if (y <= 8.5e+30) {
		tmp = fma(y, (fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), t_4);
	} else if (y <= 4e+71) {
		tmp = fma(y, fma(-1.0, (i * fma(27464.7644705, (1.0 / (y * t_3)), fma(230661.510616, (1.0 / pow(t_6, 2.0)), (t_7 / t_3)))), fma(230661.510616, (1.0 / t_6), ((27464.7644705 + (y * t_7)) / t_2))), t_4);
	} else if (y <= 1.4e+105) {
		tmp = x * ((27464.7644705 / (a * (x * y))) + ((230661.510616 / (a * (x * (y * y)))) + ((t / (a * (x * ((y * y) * y)))) + ((y / a) + (z / (a * x))))));
	} else {
		tmp = t_5;
	}
	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)
	t_2 = Float64(c + Float64(y * Float64(b + Float64(y * Float64(a + y)))))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(t / t_1)
	t_5 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	t_6 = Float64(y * t_2)
	t_7 = Float64(z + Float64(x * y))
	tmp = 0.0
	if (y <= -3.8e+47)
		tmp = t_5;
	elseif (y <= 8.5e+30)
		tmp = fma(y, Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), t_4);
	elseif (y <= 4e+71)
		tmp = fma(y, fma(-1.0, Float64(i * fma(27464.7644705, Float64(1.0 / Float64(y * t_3)), fma(230661.510616, Float64(1.0 / (t_6 ^ 2.0)), Float64(t_7 / t_3)))), fma(230661.510616, Float64(1.0 / t_6), Float64(Float64(27464.7644705 + Float64(y * t_7)) / t_2))), t_4);
	elseif (y <= 1.4e+105)
		tmp = Float64(x * Float64(Float64(27464.7644705 / Float64(a * Float64(x * y))) + Float64(Float64(230661.510616 / Float64(a * Float64(x * Float64(y * y)))) + Float64(Float64(t / Float64(a * Float64(x * Float64(Float64(y * y) * y)))) + Float64(Float64(y / a) + Float64(z / Float64(a * x)))))));
	else
		tmp = t_5;
	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]}, Block[{t$95$2 = N[(c + N[(y * N[(b + N[(y * N[(a + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, Block[{t$95$6 = N[(y * t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e+47], t$95$5, If[LessEqual[y, 8.5e+30], N[(y * N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[y, 4e+71], N[(y * N[(-1.0 * N[(i * N[(27464.7644705 * N[(1.0 / N[(y * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 * N[(1.0 / N[Power[t$95$6, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$7 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 * N[(1.0 / t$95$6), $MachinePrecision] + N[(N[(27464.7644705 + N[(y * t$95$7), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[y, 1.4e+105], N[(x * N[(N[(27464.7644705 / N[(a * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(230661.510616 / N[(a * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(a * N[(x * N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / a), $MachinePrecision] + N[(z / N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.8000000000000003e47 or 1.4000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.8000000000000003e47 < y < 8.4999999999999995e30

   1. Initial program 55.7%

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

   3. Applied rewrites 56.4%

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

   if 8.4999999999999995e30 < y < 4.0000000000000002e71

   1. Initial program 55.7%

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

   3. Applied rewrites 56.4%

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

   4. Taylor expanded in i around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1 \cdot \left(i \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{y \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \left(\frac{28832688827}{125000} \cdot \frac{1}{{y}^{2} \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \left(\frac{z}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \frac{x \cdot y}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}\right)\right)\right)\right) + \left(\frac{28832688827}{125000} \cdot \frac{1}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

   6. Applied rewrites 30.6%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(-1, i \cdot \mathsf{fma}\left(27464.7644705, \frac{1}{y \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}, \mathsf{fma}\left(230661.510616, \frac{1}{{\left(y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)\right)}^{2}}, \frac{z + x \cdot y}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}\right)\right), \mathsf{fma}\left(230661.510616, \frac{1}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}, \frac{27464.7644705 + y \cdot \left(z + x \cdot y\right)}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}\right)\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

   if 4.0000000000000002e71 < y < 1.4000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \color{blue}{\left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \color{blue}{\left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\color{blue}{\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)}} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{\color{blue}{a \cdot \left(x \cdot {y}^{3}\right)}} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \color{blue}{\left(x \cdot {y}^{3}\right)}} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(x \cdot y\right)} + \left(\frac{\frac{28832688827}{125000}}{a \cdot \left(x \cdot {y}^{2}\right)} + \left(\frac{t}{a \cdot \left(x \cdot {y}^{3}\right)} + \color{blue}{\left(\frac{y}{a} + \frac{z}{a \cdot x}\right)}\right)\right)\right) \]

   7. Applied rewrites 13.5%

      \[\leadsto x \cdot \color{blue}{\left(\frac{27464.7644705}{a \cdot \left(x \cdot y\right)} + \left(\frac{230661.510616}{a \cdot \left(x \cdot \left(y \cdot y\right)\right)} + \left(\frac{t}{a \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)} + \left(\frac{y}{a} + \frac{z}{a \cdot x}\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.9 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_1}, \frac{t}{t\_1}\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -3.8e+47)
     t_2
     (if (<= y 8.9e+55)
       (fma
        y
        (/ (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) t_1)
        (/ t t_1))
       (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_2)))))

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 t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -3.8e+47) {
		tmp = t_2;
	} else if (y <= 8.9e+55) {
		tmp = fma(y, (fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), (t / t_1));
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_2;
	}
	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)
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -3.8e+47)
		tmp = t_2;
	elseif (y <= 8.9e+55)
		tmp = fma(y, Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), Float64(t / t_1));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -3.8e+47], t$95$2, If[LessEqual[y, 8.9e+55], N[(y * N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000003e47 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.8000000000000003e47 < y < 8.9000000000000002e55

   1. Initial program 55.7%

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

   3. Applied rewrites 56.4%

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

   if 8.9000000000000002e55 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 4: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -3.8e+47)
     t_1
     (if (<= y 4e+51)
       (/
        (+
         (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
         t)
        (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
       (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -3.8e+47) {
		tmp = t_1;
	} else if (y <= 4e+51) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} 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 = -((-z - (-a * x)) / y) + x
    if (y <= (-3.8d+47)) then
        tmp = t_1
    else if (y <= 4d+51) then
        tmp = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
    else if (y <= 1.3d+105) then
        tmp = y * ((x / a) + (z / (a * y)))
    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 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -3.8e+47) {
		tmp = t_1;
	} else if (y <= 4e+51) {
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = -((-z - (-a * x)) / y) + x
	tmp = 0
	if y <= -3.8e+47:
		tmp = t_1
	elif y <= 4e+51:
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
	elif y <= 1.3e+105:
		tmp = y * ((x / a) + (z / (a * y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -3.8e+47)
		tmp = t_1;
	elseif (y <= 4e+51)
		tmp = 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));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = -((-z - (-a * x)) / y) + x;
	tmp = 0.0;
	if (y <= -3.8e+47)
		tmp = t_1;
	elseif (y <= 4e+51)
		tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	elseif (y <= 1.3e+105)
		tmp = y * ((x / a) + (z / (a * y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -3.8e+47], t$95$1, If[LessEqual[y, 4e+51], 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], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000003e47 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.8000000000000003e47 < y < 4e51

   1. Initial program 55.7%

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

   if 4e51 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 5: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, a + y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -3.8e+47)
     t_1
     (if (<= y 3e+51)
       (/
        (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
        (fma (fma (* y y) (+ a y) c) y i))
       (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -3.8e+47) {
		tmp = t_1;
	} else if (y <= 3e+51) {
		tmp = fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma((y * y), (a + y), c), y, i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -3.8e+47)
		tmp = t_1;
	elseif (y <= 3e+51)
		tmp = Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(Float64(y * y), Float64(a + y), c), y, i));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -3.8e+47], t$95$1, If[LessEqual[y, 3e+51], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(a + y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000003e47 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.8000000000000003e47 < y < 3e51

   1. Initial program 55.7%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 50.0%

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

   if 3e51 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 6: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a + y, y, b\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, x, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(y \cdot y, t\_1, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (+ a y) y b)) (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -3.1e+47)
     t_2
     (if (<= y 2e-30)
       (/
        (fma (fma (fma (* y y) x 27464.7644705) y 230661.510616) y t)
        (fma (fma t_1 y c) y i))
       (if (<= y 4.4e+51)
         (/
          (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)
          (fma (* y y) t_1 i))
         (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma((a + y), y, b);
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -3.1e+47) {
		tmp = t_2;
	} else if (y <= 2e-30) {
		tmp = fma(fma(fma((y * y), x, 27464.7644705), y, 230661.510616), y, t) / fma(fma(t_1, y, c), y, i);
	} else if (y <= 4.4e+51) {
		tmp = fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma((y * y), t_1, i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(a + y), y, b)
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -3.1e+47)
		tmp = t_2;
	elseif (y <= 2e-30)
		tmp = Float64(fma(fma(fma(Float64(y * y), x, 27464.7644705), y, 230661.510616), y, t) / fma(fma(t_1, y, c), y, i));
	elseif (y <= 4.4e+51)
		tmp = Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / fma(Float64(y * y), t_1, i));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -3.1e+47], t$95$2, If[LessEqual[y, 2e-30], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(t$95$1 * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+51], N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] * t$95$1 + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.1000000000000001e47 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.1000000000000001e47 < y < 2e-30

   1. Initial program 55.7%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + x \cdot {y}^{2}\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + x \cdot {y}^{2}\right)\right)}{\color{blue}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

   4. Applied rewrites 51.1%

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

   if 2e-30 < y < 4.39999999999999984e51

   1. Initial program 55.7%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + {y}^{2} \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i + {y}^{2} \cdot \left(b + y \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 43.3%

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

   if 4.39999999999999984e51 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 7: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0031:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, x, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{t\_1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -3.1e+47)
     t_2
     (if (<= y 0.0031)
       (/ (fma (fma (fma (* y y) x 27464.7644705) y 230661.510616) y t) t_1)
       (if (<= y 2.7e+51)
         (* (* (* y y) y) (/ z t_1))
         (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_2))))))

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 t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -3.1e+47) {
		tmp = t_2;
	} else if (y <= 0.0031) {
		tmp = fma(fma(fma((y * y), x, 27464.7644705), y, 230661.510616), y, t) / t_1;
	} else if (y <= 2.7e+51) {
		tmp = ((y * y) * y) * (z / t_1);
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_2;
	}
	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)
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -3.1e+47)
		tmp = t_2;
	elseif (y <= 0.0031)
		tmp = Float64(fma(fma(fma(Float64(y * y), x, 27464.7644705), y, 230661.510616), y, t) / t_1);
	elseif (y <= 2.7e+51)
		tmp = Float64(Float64(Float64(y * y) * y) * Float64(z / t_1));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -3.1e+47], t$95$2, If[LessEqual[y, 0.0031], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 2.7e+51], N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.1000000000000001e47 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -3.1000000000000001e47 < y < 0.00309999999999999989

   1. Initial program 55.7%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + x \cdot {y}^{2}\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + x \cdot {y}^{2}\right)\right)}{\color{blue}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

   4. Applied rewrites 51.1%

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

   if 0.00309999999999999989 < y < 2.69999999999999992e51

   1. Initial program 55.7%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 11.2%

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

   if 2.69999999999999992e51 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 8: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -5e+45)
     t_1
     (if (<= y -2.6e-11)
       (/
        (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* x y))))))
        (* a (* y y)))
       (if (<= y 7e-5)
         (/
          (fma 230661.510616 y t)
          (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
         (if (<= y 2.7e+51)
           (* (* (* y y) y) (/ z (fma (fma (fma (+ a y) y b) y c) y i)))
           (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -5e+45) {
		tmp = t_1;
	} else if (y <= -2.6e-11) {
		tmp = (230661.510616 + (y * (27464.7644705 + (y * (z + (x * y)))))) / (a * (y * y));
	} else if (y <= 7e-5) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 2.7e+51) {
		tmp = ((y * y) * y) * (z / fma(fma(fma((a + y), y, b), y, c), y, i));
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -5e+45)
		tmp = t_1;
	elseif (y <= -2.6e-11)
		tmp = Float64(Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(x * y)))))) / Float64(a * Float64(y * y)));
	elseif (y <= 7e-5)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	elseif (y <= 2.7e+51)
		tmp = Float64(Float64(Float64(y * y) * y) * Float64(z / fma(fma(fma(Float64(a + y), y, b), y, c), y, i)));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -5e+45], t$95$1, If[LessEqual[y, -2.6e-11], N[(N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e-5], N[(N[(230661.510616 * y + 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], If[LessEqual[y, 2.7e+51], N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * N[(z / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5e45 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -5e45 < y < -2.6000000000000001e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot {y}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \color{blue}{{y}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {\color{blue}{y}}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{\color{blue}{2}}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

      10. lift-*.f64 5.6

          \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

   7. Applied rewrites 5.6%

      \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot \left(y \cdot y\right)}} \]

   if -2.6000000000000001e-11 < y < 6.9999999999999994e-5

   1. Initial program 55.7%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 47.5

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

   4. Applied rewrites 47.5%

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

   if 6.9999999999999994e-5 < y < 2.69999999999999992e51

   1. Initial program 55.7%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 11.2%

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

   if 2.69999999999999992e51 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

      \[\leadsto y \cdot \color{blue}{\left(\frac{x}{a} + \frac{z}{a \cdot y}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(27464.7644705, y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -1.75e+47)
     t_1
     (if (<= y 0.00165)
       (/
        (fma (fma 27464.7644705 y 230661.510616) y t)
        (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
       (if (<= y 2.7e+51)
         (* (* (* y y) y) (/ z (fma (fma (fma (+ a y) y b) y c) y i)))
         (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -1.75e+47) {
		tmp = t_1;
	} else if (y <= 0.00165) {
		tmp = fma(fma(27464.7644705, y, 230661.510616), y, t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 2.7e+51) {
		tmp = ((y * y) * y) * (z / fma(fma(fma((a + y), y, b), y, c), y, i));
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -1.75e+47)
		tmp = t_1;
	elseif (y <= 0.00165)
		tmp = Float64(fma(fma(27464.7644705, y, 230661.510616), y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	elseif (y <= 2.7e+51)
		tmp = Float64(Float64(Float64(y * y) * y) * Float64(z / fma(fma(fma(Float64(a + y), y, b), y, c), y, i)));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -1.75e+47], t$95$1, If[LessEqual[y, 0.00165], N[(N[(N[(27464.7644705 * y + 230661.510616), $MachinePrecision] * y + 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], If[LessEqual[y, 2.7e+51], N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * N[(z / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.75000000000000008e47 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -1.75000000000000008e47 < y < 0.00165

   1. Initial program 55.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + y \cdot \left(\frac{28832688827}{125000} + \frac{54929528941}{2000000} \cdot y\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 47.5

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

   4. Applied rewrites 47.5%

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

   if 0.00165 < y < 2.69999999999999992e51

   1. Initial program 55.7%

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

   2. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) \cdot y + i} \]

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 11.2%

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

   if 2.69999999999999992e51 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 10: 73.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(y \cdot y\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{t\_1}\\ \mathbf{elif}\;y \leq 0.0031:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{27464.7644705}{t\_1} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* y y))) (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -5e+45)
     t_2
     (if (<= y -2.6e-11)
       (/ (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* x y)))))) t_1)
       (if (<= y 0.0031)
         (/
          (fma 230661.510616 y t)
          (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
         (if (<= y 1.3e+105)
           (* y (+ (/ 27464.7644705 t_1) (+ (/ x a) (/ z (* a y)))))
           t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a * (y * y);
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -5e+45) {
		tmp = t_2;
	} else if (y <= -2.6e-11) {
		tmp = (230661.510616 + (y * (27464.7644705 + (y * (z + (x * y)))))) / t_1;
	} else if (y <= 0.0031) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((27464.7644705 / t_1) + ((x / a) + (z / (a * y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a * Float64(y * y))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -5e+45)
		tmp = t_2;
	elseif (y <= -2.6e-11)
		tmp = Float64(Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(x * y)))))) / t_1);
	elseif (y <= 0.0031)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(27464.7644705 / t_1) + Float64(Float64(x / a) + Float64(z / Float64(a * y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -5e+45], t$95$2, If[LessEqual[y, -2.6e-11], N[(N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 0.0031], N[(N[(230661.510616 * y + 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], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(27464.7644705 / t$95$1), $MachinePrecision] + N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5e45 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -5e45 < y < -2.6000000000000001e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot {y}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \color{blue}{{y}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {\color{blue}{y}}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{\color{blue}{2}}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

      10. lift-*.f64 5.6

          \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

   7. Applied rewrites 5.6%

      \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot \left(y \cdot y\right)}} \]

   if -2.6000000000000001e-11 < y < 0.00309999999999999989

   1. Initial program 55.7%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 47.5

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

   4. Applied rewrites 47.5%

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

   if 0.00309999999999999989 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \color{blue}{\left(\frac{x}{a} + \frac{z}{a \cdot y}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \color{blue}{\frac{z}{a \cdot y}}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \frac{\color{blue}{z}}{a \cdot y}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

      5. pow2 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 10.7

          \[\leadsto y \cdot \left(\frac{27464.7644705}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

   7. Applied rewrites 10.7%

      \[\leadsto y \cdot \color{blue}{\left(\frac{27464.7644705}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(y \cdot y\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{t\_1}\\ \mathbf{elif}\;y \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, a + y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{27464.7644705}{t\_1} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* a (* y y))) (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -5e+45)
     t_2
     (if (<= y -2.6e-11)
       (/ (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* x y)))))) t_1)
       (if (<= y 0.00165)
         (/ (fma 230661.510616 y t) (fma (fma (* y y) (+ a y) c) y i))
         (if (<= y 1.3e+105)
           (* y (+ (/ 27464.7644705 t_1) (+ (/ x a) (/ z (* a y)))))
           t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = a * (y * y);
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -5e+45) {
		tmp = t_2;
	} else if (y <= -2.6e-11) {
		tmp = (230661.510616 + (y * (27464.7644705 + (y * (z + (x * y)))))) / t_1;
	} else if (y <= 0.00165) {
		tmp = fma(230661.510616, y, t) / fma(fma((y * y), (a + y), c), y, i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((27464.7644705 / t_1) + ((x / a) + (z / (a * y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a * Float64(y * y))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -5e+45)
		tmp = t_2;
	elseif (y <= -2.6e-11)
		tmp = Float64(Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(x * y)))))) / t_1);
	elseif (y <= 0.00165)
		tmp = Float64(fma(230661.510616, y, t) / fma(fma(Float64(y * y), Float64(a + y), c), y, i));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(27464.7644705 / t_1) + Float64(Float64(x / a) + Float64(z / Float64(a * y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a * N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -5e+45], t$95$2, If[LessEqual[y, -2.6e-11], N[(N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 0.00165], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(a + y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(27464.7644705 / t$95$1), $MachinePrecision] + N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5e45 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -5e45 < y < -2.6000000000000001e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot {y}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \color{blue}{{y}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {\color{blue}{y}}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{\color{blue}{2}}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

      10. lift-*.f64 5.6

          \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

   7. Applied rewrites 5.6%

      \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot \left(y \cdot y\right)}} \]

   if -2.6000000000000001e-11 < y < 0.00165

   1. Initial program 55.7%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 43.9%

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

   if 0.00165 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto y \cdot \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \color{blue}{\left(\frac{x}{a} + \frac{z}{a \cdot y}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \color{blue}{\frac{z}{a \cdot y}}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \frac{\color{blue}{z}}{a \cdot y}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot {y}^{2}} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

      5. pow2 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto y \cdot \left(\frac{\frac{54929528941}{2000000}}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 10.7

          \[\leadsto y \cdot \left(\frac{27464.7644705}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right) \]

   7. Applied rewrites 10.7%

      \[\leadsto y \cdot \color{blue}{\left(\frac{27464.7644705}{a \cdot \left(y \cdot y\right)} + \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, a + y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -5e+45)
     t_1
     (if (<= y -2.6e-11)
       (/
        (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* x y))))))
        (* a (* y y)))
       (if (<= y 0.00165)
         (/ (fma 230661.510616 y t) (fma (fma (* y y) (+ a y) c) y i))
         (if (<= y 1.3e+105) (* y (+ (/ x a) (/ z (* a y)))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -5e+45) {
		tmp = t_1;
	} else if (y <= -2.6e-11) {
		tmp = (230661.510616 + (y * (27464.7644705 + (y * (z + (x * y)))))) / (a * (y * y));
	} else if (y <= 0.00165) {
		tmp = fma(230661.510616, y, t) / fma(fma((y * y), (a + y), c), y, i);
	} else if (y <= 1.3e+105) {
		tmp = y * ((x / a) + (z / (a * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -5e+45)
		tmp = t_1;
	elseif (y <= -2.6e-11)
		tmp = Float64(Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(x * y)))))) / Float64(a * Float64(y * y)));
	elseif (y <= 0.00165)
		tmp = Float64(fma(230661.510616, y, t) / fma(fma(Float64(y * y), Float64(a + y), c), y, i));
	elseif (y <= 1.3e+105)
		tmp = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -5e+45], t$95$1, If[LessEqual[y, -2.6e-11], N[(N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00165], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(a + y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5e45 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -5e45 < y < -2.6000000000000001e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot {y}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \color{blue}{{y}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {\color{blue}{y}}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot {y}^{\color{blue}{2}}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

      10. lift-*.f64 5.6

          \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{a \cdot \left(y \cdot y\right)} \]

   7. Applied rewrites 5.6%

      \[\leadsto \frac{230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right)}{\color{blue}{a \cdot \left(y \cdot y\right)}} \]

   if -2.6000000000000001e-11 < y < 0.00165

   1. Initial program 55.7%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 43.9%

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

   if 0.00165 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

Alternative 13: 69.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, a + y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* y (+ (/ x a) (/ z (* a y)))))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -8.5e+46)
     t_2
     (if (<= y -2.6e-11)
       t_1
       (if (<= y 0.00165)
         (/ (fma 230661.510616 y t) (fma (fma (* y y) (+ a y) c) y i))
         (if (<= y 1.3e+105) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * ((x / a) + (z / (a * y)));
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -8.5e+46) {
		tmp = t_2;
	} else if (y <= -2.6e-11) {
		tmp = t_1;
	} else if (y <= 0.00165) {
		tmp = fma(230661.510616, y, t) / fma(fma((y * y), (a + y), c), y, i);
	} else if (y <= 1.3e+105) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -8.5e+46)
		tmp = t_2;
	elseif (y <= -2.6e-11)
		tmp = t_1;
	elseif (y <= 0.00165)
		tmp = Float64(fma(230661.510616, y, t) / fma(fma(Float64(y * y), Float64(a + y), c), y, i));
	elseif (y <= 1.3e+105)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -8.5e+46], t$95$2, If[LessEqual[y, -2.6e-11], t$95$1, If[LessEqual[y, 0.00165], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(a + y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999996e46 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -8.4999999999999996e46 < y < -2.6000000000000001e-11 or 0.00165 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -2.6000000000000001e-11 < y < 0.00165

   1. Initial program 55.7%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 43.9%

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

Alternative 14: 66.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0031:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* y (+ (/ x a) (/ z (* a y)))))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -8.5e+46)
     t_2
     (if (<= y -2.25e-11)
       t_1
       (if (<= y 0.0031)
         (/ t (fma (fma (fma (+ a y) y b) y c) y i))
         (if (<= y 1.3e+105) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * ((x / a) + (z / (a * y)));
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -8.5e+46) {
		tmp = t_2;
	} else if (y <= -2.25e-11) {
		tmp = t_1;
	} else if (y <= 0.0031) {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else if (y <= 1.3e+105) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -8.5e+46)
		tmp = t_2;
	elseif (y <= -2.25e-11)
		tmp = t_1;
	elseif (y <= 0.0031)
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	elseif (y <= 1.3e+105)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -8.5e+46], t$95$2, If[LessEqual[y, -2.25e-11], t$95$1, If[LessEqual[y, 0.0031], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999996e46 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -8.4999999999999996e46 < y < -2.25e-11 or 0.00309999999999999989 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -2.25e-11 < y < 0.00309999999999999989

   1. Initial program 55.7%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 40.6

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

   4. Applied rewrites 40.6%

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

Alternative 15: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00165:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, a + y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* y (+ (/ x a) (/ z (* a y)))))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -8.5e+46)
     t_2
     (if (<= y -2.25e-11)
       t_1
       (if (<= y 0.00165)
         (/ t (fma (fma (* y y) (+ a y) c) y i))
         (if (<= y 1.3e+105) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * ((x / a) + (z / (a * y)));
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -8.5e+46) {
		tmp = t_2;
	} else if (y <= -2.25e-11) {
		tmp = t_1;
	} else if (y <= 0.00165) {
		tmp = t / fma(fma((y * y), (a + y), c), y, i);
	} else if (y <= 1.3e+105) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -8.5e+46)
		tmp = t_2;
	elseif (y <= -2.25e-11)
		tmp = t_1;
	elseif (y <= 0.00165)
		tmp = Float64(t / fma(fma(Float64(y * y), Float64(a + y), c), y, i));
	elseif (y <= 1.3e+105)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -8.5e+46], t$95$2, If[LessEqual[y, -2.25e-11], t$95$1, If[LessEqual[y, 0.00165], N[(t / N[(N[(N[(y * y), $MachinePrecision] * N[(a + y), $MachinePrecision] + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999996e46 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -8.4999999999999996e46 < y < -2.25e-11 or 0.00165 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -2.25e-11 < y < 0.00165

   1. Initial program 55.7%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{i + y \cdot \left(c + {y}^{2} \cdot \left(a + y\right)\right)}} \]

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 37.9%

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

Alternative 16: 60.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot z, y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* y (+ (/ x a) (/ z (* a y)))))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -8.5e+46)
     t_2
     (if (<= y -2.55e-11)
       t_1
       (if (<= y 9.5e-10)
         (/ (fma (fma (* y z) y 230661.510616) y t) i)
         (if (<= y 1.3e+105) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * ((x / a) + (z / (a * y)));
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -8.5e+46) {
		tmp = t_2;
	} else if (y <= -2.55e-11) {
		tmp = t_1;
	} else if (y <= 9.5e-10) {
		tmp = fma(fma((y * z), y, 230661.510616), y, t) / i;
	} else if (y <= 1.3e+105) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -8.5e+46)
		tmp = t_2;
	elseif (y <= -2.55e-11)
		tmp = t_1;
	elseif (y <= 9.5e-10)
		tmp = Float64(fma(fma(Float64(y * z), y, 230661.510616), y, t) / i);
	elseif (y <= 1.3e+105)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -8.5e+46], t$95$2, If[LessEqual[y, -2.55e-11], t$95$1, If[LessEqual[y, 9.5e-10], N[(N[(N[(N[(y * z), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[y, 1.3e+105], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999996e46 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -8.4999999999999996e46 < y < -2.54999999999999992e-11 or 9.50000000000000028e-10 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -2.54999999999999992e-11 < y < 9.50000000000000028e-10

   1. Initial program 55.7%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot z, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 32.9

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot z, y, 230661.510616\right), y, t\right)}{i} \]

   7. Applied rewrites 32.9%

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

Alternative 17: 59.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\frac{x}{a} + \frac{z}{a \cdot y}\right)\\ t_2 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(230661.510616, \frac{y}{i}, \frac{t}{i}\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* y (+ (/ x a) (/ z (* a y)))))
        (t_2 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -8.5e+46)
     t_2
     (if (<= y -1.3e-11)
       t_1
       (if (<= y 1.95e-43)
         (fma 230661.510616 (/ y i) (/ t i))
         (if (<= y 1.3e+105) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * ((x / a) + (z / (a * y)));
	double t_2 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -8.5e+46) {
		tmp = t_2;
	} else if (y <= -1.3e-11) {
		tmp = t_1;
	} else if (y <= 1.95e-43) {
		tmp = fma(230661.510616, (y / i), (t / i));
	} else if (y <= 1.3e+105) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(y * Float64(Float64(x / a) + Float64(z / Float64(a * y))))
	t_2 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -8.5e+46)
		tmp = t_2;
	elseif (y <= -1.3e-11)
		tmp = t_1;
	elseif (y <= 1.95e-43)
		tmp = fma(230661.510616, Float64(y / i), Float64(t / i));
	elseif (y <= 1.3e+105)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(N[(x / a), $MachinePrecision] + N[(z / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -8.5e+46], t$95$2, If[LessEqual[y, -1.3e-11], t$95$1, If[LessEqual[y, 1.95e-43], N[(230661.510616 * N[(y / i), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+105], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999996e46 or 1.3000000000000001e105 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -8.4999999999999996e46 < y < -1.3e-11 or 1.95e-43 < y < 1.3000000000000001e105

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -1.3e-11 < y < 1.95e-43

   1. Initial program 55.7%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{28832688827}{125000}, \frac{y}{i}, \frac{t}{i}\right) \]

      3. lower-/.f64 31.7

         \[\leadsto \mathsf{fma}\left(230661.510616, \frac{y}{i}, \frac{t}{i}\right) \]

   7. Applied rewrites 31.7%

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

Alternative 18: 58.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\frac{\left(-z\right) - \left(-a\right) \cdot x}{y}\right) + x\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-44}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(27464.7644705, y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (- (/ (- (- z) (* (- a) x)) y)) x)))
   (if (<= y -2.9e+46)
     t_1
     (if (<= y -6e-44)
       (/ (* x y) a)
       (if (<= y 1.25e+28)
         (/ (fma (fma 27464.7644705 y 230661.510616) y t) 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 = -((-z - (-a * x)) / y) + x;
	double tmp;
	if (y <= -2.9e+46) {
		tmp = t_1;
	} else if (y <= -6e-44) {
		tmp = (x * y) / a;
	} else if (y <= 1.25e+28) {
		tmp = fma(fma(27464.7644705, y, 230661.510616), y, t) / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-Float64(Float64(Float64(-z) - Float64(Float64(-a) * x)) / y)) + x)
	tmp = 0.0
	if (y <= -2.9e+46)
		tmp = t_1;
	elseif (y <= -6e-44)
		tmp = Float64(Float64(x * y) / a);
	elseif (y <= 1.25e+28)
		tmp = Float64(fma(fma(27464.7644705, y, 230661.510616), y, t) / i);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-N[(N[((-z) - N[((-a) * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]) + x), $MachinePrecision]}, If[LessEqual[y, -2.9e+46], t$95$1, If[LessEqual[y, -6e-44], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 1.25e+28], N[(N[(N[(27464.7644705 * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9000000000000002e46 or 1.24999999999999989e28 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 31.3

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

   4. Applied rewrites 31.3%

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

   if -2.9000000000000002e46 < y < -6.0000000000000005e-44

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{a} \]

      2. lift-*.f64 9.5

         \[\leadsto \frac{x \cdot y}{a} \]

   7. Applied rewrites 9.5%

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

   if -6.0000000000000005e-44 < y < 1.24999999999999989e28

   1. Initial program 55.7%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

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

Alternative 19: 35.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{a}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(27464.7644705, y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.3e-11)
   (/ z a)
   (if (<= y 3.2e+56)
     (/ (fma (fma 27464.7644705 y 230661.510616) y t) i)
     (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.3e-11) {
		tmp = z / a;
	} else if (y <= 3.2e+56) {
		tmp = fma(fma(27464.7644705, y, 230661.510616), y, t) / i;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.3e-11)
		tmp = Float64(z / a);
	elseif (y <= 3.2e+56)
		tmp = Float64(fma(fma(27464.7644705, y, 230661.510616), y, t) / i);
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.3e-11], N[(z / a), $MachinePrecision], If[LessEqual[y, 3.2e+56], N[(N[(N[(27464.7644705 * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / i), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 7.9

         \[\leadsto \frac{z}{a} \]

   7. Applied rewrites 7.9%

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

   if -1.3e-11 < y < 3.20000000000000003e56

   1. Initial program 55.7%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{54929528941}{2000000}, y, \frac{28832688827}{125000}\right), y, t\right)}{i} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(27464.7644705, y, 230661.510616\right), y, t\right)}{i} \]

   if 3.20000000000000003e56 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{a} \]

      2. lift-*.f64 9.5

         \[\leadsto \frac{x \cdot y}{a} \]

   7. Applied rewrites 9.5%

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

Alternative 20: 35.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{a}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(230661.510616, \frac{y}{i}, \frac{t}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.3e-11)
   (/ z a)
   (if (<= y 2.45e-43) (fma 230661.510616 (/ y i) (/ t i)) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.3e-11) {
		tmp = z / a;
	} else if (y <= 2.45e-43) {
		tmp = fma(230661.510616, (y / i), (t / i));
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.3e-11)
		tmp = Float64(z / a);
	elseif (y <= 2.45e-43)
		tmp = fma(230661.510616, Float64(y / i), Float64(t / i));
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.3e-11], N[(z / a), $MachinePrecision], If[LessEqual[y, 2.45e-43], N[(230661.510616 * N[(y / i), $MachinePrecision] + N[(t / i), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 7.9

         \[\leadsto \frac{z}{a} \]

   7. Applied rewrites 7.9%

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

   if -1.3e-11 < y < 2.44999999999999994e-43

   1. Initial program 55.7%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{28832688827}{125000}, \frac{y}{i}, \frac{t}{i}\right) \]

      3. lower-/.f64 31.7

         \[\leadsto \mathsf{fma}\left(230661.510616, \frac{y}{i}, \frac{t}{i}\right) \]

   7. Applied rewrites 31.7%

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

   if 2.44999999999999994e-43 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{a} \]

      2. lift-*.f64 9.5

         \[\leadsto \frac{x \cdot y}{a} \]

   7. Applied rewrites 9.5%

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

Alternative 21: 35.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{a}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.3e-11)
   (/ z a)
   (if (<= y 2.45e-43) (/ (fma 230661.510616 y t) i) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.3e-11) {
		tmp = z / a;
	} else if (y <= 2.45e-43) {
		tmp = fma(230661.510616, y, t) / i;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.3e-11)
		tmp = Float64(z / a);
	elseif (y <= 2.45e-43)
		tmp = Float64(fma(230661.510616, y, t) / i);
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.3e-11], N[(z / a), $MachinePrecision], If[LessEqual[y, 2.45e-43], N[(N[(230661.510616 * y + t), $MachinePrecision] / i), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 7.9

         \[\leadsto \frac{z}{a} \]

   7. Applied rewrites 7.9%

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

   if -1.3e-11 < y < 2.44999999999999994e-43

   1. Initial program 55.7%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{28832688827}{125000}, y, t\right)}{i} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto \frac{\mathsf{fma}\left(230661.510616, y, t\right)}{i} \]

   if 2.44999999999999994e-43 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{a} \]

      2. lift-*.f64 9.5

         \[\leadsto \frac{x \cdot y}{a} \]

   7. Applied rewrites 9.5%

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

Alternative 22: 32.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{a}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+45}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -4.5e-11) (/ z a) (if (<= y 5.3e+45) (/ t i) (/ (* x y) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -4.5e-11) {
		tmp = z / a;
	} else if (y <= 5.3e+45) {
		tmp = t / i;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-4.5d-11)) then
        tmp = z / a
    else if (y <= 5.3d+45) then
        tmp = t / i
    else
        tmp = (x * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -4.5e-11) {
		tmp = z / a;
	} else if (y <= 5.3e+45) {
		tmp = t / i;
	} else {
		tmp = (x * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -4.5e-11:
		tmp = z / a
	elif y <= 5.3e+45:
		tmp = t / i
	else:
		tmp = (x * y) / a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -4.5e-11)
		tmp = Float64(z / a);
	elseif (y <= 5.3e+45)
		tmp = Float64(t / i);
	else
		tmp = Float64(Float64(x * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -4.5e-11)
		tmp = z / a;
	elseif (y <= 5.3e+45)
		tmp = t / i;
	else
		tmp = (x * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -4.5e-11], N[(z / a), $MachinePrecision], If[LessEqual[y, 5.3e+45], N[(t / i), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 7.9

         \[\leadsto \frac{z}{a} \]

   7. Applied rewrites 7.9%

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

   if -4.5e-11 < y < 5.29999999999999991e45

   1. Initial program 55.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 28.6

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

   4. Applied rewrites 28.6%

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

   if 5.29999999999999991e45 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{a} \]

      2. lift-*.f64 9.5

         \[\leadsto \frac{x \cdot y}{a} \]

   7. Applied rewrites 9.5%

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

Alternative 23: 31.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{z}{a}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -4.5e-11) (/ z a) (if (<= y 2.7e-11) (/ t i) (/ z a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -4.5e-11) {
		tmp = z / a;
	} else if (y <= 2.7e-11) {
		tmp = t / i;
	} else {
		tmp = z / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-4.5d-11)) then
        tmp = z / a
    else if (y <= 2.7d-11) then
        tmp = t / i
    else
        tmp = z / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -4.5e-11) {
		tmp = z / a;
	} else if (y <= 2.7e-11) {
		tmp = t / i;
	} else {
		tmp = z / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -4.5e-11:
		tmp = z / a
	elif y <= 2.7e-11:
		tmp = t / i
	else:
		tmp = z / a
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -4.5e-11)
		tmp = Float64(z / a);
	elseif (y <= 2.7e-11)
		tmp = Float64(t / i);
	else
		tmp = Float64(z / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -4.5e-11)
		tmp = z / a;
	elseif (y <= 2.7e-11)
		tmp = t / i;
	else
		tmp = z / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -4.5e-11], N[(z / a), $MachinePrecision], If[LessEqual[y, 2.7e-11], N[(t / i), $MachinePrecision], N[(z / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e-11 or 2.70000000000000005e-11 < y

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

   4. Applied rewrites 6.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \frac{z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 7.9

         \[\leadsto \frac{z}{a} \]

   7. Applied rewrites 7.9%

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

   if -4.5e-11 < y < 2.70000000000000005e-11

   1. Initial program 55.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 28.6

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

   4. Applied rewrites 28.6%

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

Alternative 24: 7.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{z}{a} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = z / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{a \cdot {y}^{3}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{\color{blue}{a \cdot {y}^{3}}} \]

4. Applied rewrites 6.3%

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

5. Taylor expanded in z around inf

   \[\leadsto \frac{z}{\color{blue}{a}} \]
6. Step-by-step derivation

   1. lower-/.f64 7.9

      \[\leadsto \frac{z}{a} \]

7. Applied rewrites 7.9%

   \[\leadsto \frac{z}{\color{blue}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(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)))