AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Percentage Accurate: 61.8% → 90.4%

Time: 5.6s

Alternatives: 15

Speedup: 1.5×

Specification

Math

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

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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
    code = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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
    code = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(a + z\right) - b \cdot \frac{y}{x + y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y + x}{t\_1}, \frac{a \cdot \left(t + y\right) - b \cdot y}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{y + x}, y, z - \frac{y}{y + x} \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (+ (+ t x) y))
       (t_2
        (/
         (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b))
         (+ (+ x t) y))))
  (if (<= t_2 (- INFINITY))
    (- (+ a z) (* b (/ y (+ x y))))
    (if (<= t_2 5e+289)
      (fma z (/ (+ y x) t_1) (/ (- (* a (+ t y)) (* b y)) t_1))
      (fma (/ a (+ y x)) y (- z (* (/ y (+ y x)) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (a + z) - (b * (y / (x + y)));
	} else if (t_2 <= 5e+289) {
		tmp = fma(z, ((y + x) / t_1), (((a * (t + y)) - (b * y)) / t_1));
	} else {
		tmp = fma((a / (y + x)), y, (z - ((y / (y + x)) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))));
	elseif (t_2 <= 5e+289)
		tmp = fma(z, Float64(Float64(y + x) / t_1), Float64(Float64(Float64(a * Float64(t + y)) - Float64(b * y)) / t_1));
	else
		tmp = fma(Float64(a / Float64(y + x)), y, Float64(z - Float64(Float64(y / Float64(y + x)) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+289], N[(z * N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(a * N[(t + y), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(y + x), $MachinePrecision]), $MachinePrecision] * y + N[(z - N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000003e289

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 73.6%

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

   if 5.0000000000000003e289 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 65.1%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 65.1%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 65.1%

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

   8. Applied rewrites 65.1%

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

Alternative 2: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(a + z\right) - b \cdot \frac{y}{x + y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(x, z, y \cdot \left(\left(a + z\right) - b\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{y + x}, y, z - \frac{y}{y + x} \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (+ (+ x t) y))
       (t_2 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) t_1)))
  (if (<= t_2 (- INFINITY))
    (- (+ a z) (* b (/ y (+ x y))))
    (if (<= t_2 5e+289)
      (/ (fma a t (fma x z (* y (- (+ a z) b)))) t_1)
      (fma (/ a (+ y x)) y (- z (* (/ y (+ y x)) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (a + z) - (b * (y / (x + y)));
	} else if (t_2 <= 5e+289) {
		tmp = fma(a, t, fma(x, z, (y * ((a + z) - b)))) / t_1;
	} else {
		tmp = fma((a / (y + x)), y, (z - ((y / (y + x)) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))));
	elseif (t_2 <= 5e+289)
		tmp = Float64(fma(a, t, fma(x, z, Float64(y * Float64(Float64(a + z) - b)))) / t_1);
	else
		tmp = fma(Float64(a / Float64(y + x)), y, Float64(z - Float64(Float64(y / Float64(y + x)) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+289], N[(N[(a * t + N[(x * z + N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(a / N[(y + x), $MachinePrecision]), $MachinePrecision] * y + N[(z - N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000003e289

   1. Initial program 61.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 62.0%

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

   4. Applied rewrites 62.0%

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

   if 5.0000000000000003e289 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 65.1%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 65.1%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 65.1%

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

   8. Applied rewrites 65.1%

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

Alternative 3: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+185}:\\ \;\;\;\;\left(a + z\right) - b \cdot \frac{y}{x + y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{y + x}, y, z - \frac{y}{y + x} \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1
        (/
         (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b))
         (+ (+ x t) y))))
  (if (<= t_1 -2e+185)
    (- (+ a z) (* b (/ y (+ x y))))
    (if (<= t_1 5e+289)
      (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y)))
      (fma (/ a (+ y x)) y (- z (* (/ y (+ y x)) b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if (t_1 <= -2e+185) {
		tmp = (a + z) - (b * (y / (x + y)));
	} else if (t_1 <= 5e+289) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else {
		tmp = fma((a / (y + x)), y, (z - ((y / (y + x)) * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	tmp = 0.0
	if (t_1 <= -2e+185)
		tmp = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))));
	elseif (t_1 <= 5e+289)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	else
		tmp = fma(Float64(a / Float64(y + x)), y, Float64(z - Float64(Float64(y / Float64(y + x)) * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+185], N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+289], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(y + x), $MachinePrecision]), $MachinePrecision] * y + N[(z - N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2e185

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

   if -2e185 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000003e289

   1. Initial program 61.8%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 48.7%

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

   4. Applied rewrites 48.7%

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

   if 5.0000000000000003e289 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 65.1%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 65.1%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 65.1%

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

   8. Applied rewrites 65.1%

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

Alternative 4: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(a + z\right) - b \cdot \frac{y}{x + y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+185}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+187}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1
        (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
       (t_2 (- (+ a z) (* b (/ y (+ x y))))))
  (if (<= t_1 -2e+185)
    t_2
    (if (<= t_1 1e+187)
      (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y)))
      t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - (b * (y / (x + y)));
	double tmp;
	if (t_1 <= -2e+185) {
		tmp = t_2;
	} else if (t_1 <= 1e+187) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))))
	tmp = 0.0
	if (t_1 <= -2e+185)
		tmp = t_2;
	elseif (t_1 <= 1e+187)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+185], t$95$2, If[LessEqual[t$95$1, 1e+187], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2e185 or 9.9999999999999991e186 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

   if -2e185 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999991e186

   1. Initial program 61.8%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 48.7%

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

   4. Applied rewrites 48.7%

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

Alternative 5: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{y + x}, y, z - b\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right) - b \cdot y}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b \cdot \frac{y}{x + y}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= y -2.2e+92)
  (fma (/ a (+ y x)) y (- z b))
  (if (<= y 2.25e-93)
    (fma a (/ (+ t y) (+ (+ t x) y)) z)
    (if (<= y 2.5e+43)
      (/ (- (* z (+ x y)) (* b y)) (+ (+ x t) y))
      (- (+ a z) (* b (/ y (+ x y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.2e+92) {
		tmp = fma((a / (y + x)), y, (z - b));
	} else if (y <= 2.25e-93) {
		tmp = fma(a, ((t + y) / ((t + x) + y)), z);
	} else if (y <= 2.5e+43) {
		tmp = ((z * (x + y)) - (b * y)) / ((x + t) + y);
	} else {
		tmp = (a + z) - (b * (y / (x + y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.2e+92)
		tmp = fma(Float64(a / Float64(y + x)), y, Float64(z - b));
	elseif (y <= 2.25e-93)
		tmp = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), z);
	elseif (y <= 2.5e+43)
		tmp = Float64(Float64(Float64(z * Float64(x + y)) - Float64(b * y)) / Float64(Float64(x + t) + y));
	else
		tmp = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.2e+92], N[(N[(a / N[(y + x), $MachinePrecision]), $MachinePrecision] * y + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-93], N[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[y, 2.5e+43], N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.1999999999999999e92

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 65.1%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 65.1%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 65.1%

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{a}{y + x}, y, z - b\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.4%

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

   if -2.1999999999999999e92 < y < 2.2500000000000001e-93

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. div-add N/A

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

      7. sub-flip-reverse N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 73.3%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 65.2%

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

   if 2.2500000000000001e-93 < y < 2.5000000000000002e43

   1. Initial program 61.8%

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

   2. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 38.8%

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

   4. Applied rewrites 38.8%

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

   if 2.5000000000000002e43 < y

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

Alternative 6: 69.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z\right)\\ \mathbf{if}\;a \leq -5.3 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-58}:\\ \;\;\;\;z - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (fma a (/ (+ t y) (+ (+ t x) y)) z)))
  (if (<= a -5.3e+16)
    t_1
    (if (<= a 1.5e-58) (- z (* b (/ y (+ (+ x t) y)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, ((t + y) / ((t + x) + y)), z);
	double tmp;
	if (a <= -5.3e+16) {
		tmp = t_1;
	} else if (a <= 1.5e-58) {
		tmp = z - (b * (y / ((x + t) + y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), z)
	tmp = 0.0
	if (a <= -5.3e+16)
		tmp = t_1;
	elseif (a <= 1.5e-58)
		tmp = Float64(z - Float64(b * Float64(y / Float64(Float64(x + t) + y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[a, -5.3e+16], t$95$1, If[LessEqual[a, 1.5e-58], N[(z - N[(b * N[(y / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.3e16 or 1.5e-58 < a

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. div-add N/A

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

      7. sub-flip-reverse N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 73.3%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 65.2%

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

   if -5.3e16 < a < 1.5e-58

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. div-add N/A

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

      7. sub-flip-reverse N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.0%

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

Alternative 7: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+82}:\\ \;\;\;\;z - b \cdot \frac{y}{\left(x + t\right) + y}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+199}:\\ \;\;\;\;\left(a + z\right) - b \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a - b}{x}, y, z\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= x -8.6e+82)
  (- z (* b (/ y (+ (+ x t) y))))
  (if (<= x 4.5e+199)
    (- (+ a z) (* b (/ y (+ x y))))
    (fma (/ (- a b) x) y z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -8.6e+82) {
		tmp = z - (b * (y / ((x + t) + y)));
	} else if (x <= 4.5e+199) {
		tmp = (a + z) - (b * (y / (x + y)));
	} else {
		tmp = fma(((a - b) / x), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -8.6e+82)
		tmp = Float64(z - Float64(b * Float64(y / Float64(Float64(x + t) + y))));
	elseif (x <= 4.5e+199)
		tmp = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))));
	else
		tmp = fma(Float64(Float64(a - b) / x), y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -8.6e+82], N[(z - N[(b * N[(y / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e+199], N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - b), $MachinePrecision] / x), $MachinePrecision] * y + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.6000000000000003e82

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. div-add N/A

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

      7. sub-flip-reverse N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 73.3%

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 52.0%

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

   if -8.6000000000000003e82 < x < 4.4999999999999997e199

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

   if 4.4999999999999997e199 < x

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 34.0%

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

   7. Applied rewrites 34.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      9. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

      11. lower--.f64 34.1%

          \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

   9. Applied rewrites 34.1%

      \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(a + z\right) - b \cdot \frac{y}{x + y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+58}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1
        (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
       (t_2 (- (+ a z) (* b (/ y (+ x y))))))
  (if (<= t_1 -2e+58)
    t_2
    (if (<= t_1 1e+140) (/ (fma a t (* x z)) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (a + z) - (b * (y / (x + y)));
	double tmp;
	if (t_1 <= -2e+58) {
		tmp = t_2;
	} else if (t_1 <= 1e+140) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(a + z) - Float64(b * Float64(y / Float64(x + y))))
	tmp = 0.0
	if (t_1 <= -2e+58)
		tmp = t_2;
	elseif (t_1 <= 1e+140)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - N[(b * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+58], t$95$2, If[LessEqual[t$95$1, 1e+140], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999999e58 or 1.0000000000000001e140 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 63.6%

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

   9. Applied rewrites 63.6%

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

   if -1.9999999999999999e58 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e140

   1. Initial program 61.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 42.0%

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{y + x}, y, z - b\right)\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-193}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+48}:\\ \;\;\;\;z - \frac{b \cdot y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= y -2.2e-59)
  (fma (/ a (+ y x)) y (- z b))
  (if (<= y 1.08e-193)
    (/ (fma a t (* x z)) (+ t x))
    (if (<= y 4.2e+48) (- z (/ (* b y) (+ x y))) (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.2e-59) {
		tmp = fma((a / (y + x)), y, (z - b));
	} else if (y <= 1.08e-193) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (y <= 4.2e+48) {
		tmp = z - ((b * y) / (x + y));
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.2e-59)
		tmp = fma(Float64(a / Float64(y + x)), y, Float64(z - b));
	elseif (y <= 1.08e-193)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (y <= 4.2e+48)
		tmp = Float64(z - Float64(Float64(b * y) / Float64(x + y)));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.2e-59], N[(N[(a / N[(y + x), $MachinePrecision]), $MachinePrecision] * y + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.08e-193], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+48], N[(z - N[(N[(b * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.1999999999999999e-59

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 65.1%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 65.1%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 65.1%

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

   8. Applied rewrites 65.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{a}{y + x}, y, z - b\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.4%

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

   if -2.1999999999999999e-59 < y < 1.08e-193

   1. Initial program 61.8%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 42.0%

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

   4. Applied rewrites 42.0%

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

   if 1.08e-193 < y < 4.1999999999999997e48

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. add-to-fraction-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.7%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 40.7%

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

   9. Applied rewrites 40.7%

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

   if 4.1999999999999997e48 < y

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 54.9%

         \[\leadsto \left(a + z\right) - b \]

   4. Applied rewrites 54.9%

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

Alternative 10: 62.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a - b}{x}, y, z\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+190}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (let* ((t_1 (fma (/ (- a b) x) y z)))
  (if (<= x -3.4e+134) t_1 (if (<= x 2.2e+190) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((a - b) / x), y, z);
	double tmp;
	if (x <= -3.4e+134) {
		tmp = t_1;
	} else if (x <= 2.2e+190) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(a - b) / x), y, z)
	tmp = 0.0
	if (x <= -3.4e+134)
		tmp = t_1;
	elseif (x <= 2.2e+190)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a - b), $MachinePrecision] / x), $MachinePrecision] * y + z), $MachinePrecision]}, If[LessEqual[x, -3.4e+134], t$95$1, If[LessEqual[x, 2.2e+190], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4000000000000002e134 or 2.2e190 < x

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 34.0%

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

   7. Applied rewrites 34.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      9. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

      11. lower--.f64 34.1%

          \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

   9. Applied rewrites 34.1%

      \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

   if -3.4000000000000002e134 < x < 2.2e190

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 54.9%

         \[\leadsto \left(a + z\right) - b \]

   4. Applied rewrites 54.9%

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

Alternative 11: 58.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+183}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z + \frac{a \cdot y}{x}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= x -1.32e+147)
  z
  (if (<= x 4.2e+183) (- (+ a z) b) (+ z (/ (* a y) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.32e+147) {
		tmp = z;
	} else if (x <= 4.2e+183) {
		tmp = (a + z) - b;
	} else {
		tmp = z + ((a * y) / x);
	}
	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)
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) :: tmp
    if (x <= (-1.32d+147)) then
        tmp = z
    else if (x <= 4.2d+183) then
        tmp = (a + z) - b
    else
        tmp = z + ((a * y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.32e+147) {
		tmp = z;
	} else if (x <= 4.2e+183) {
		tmp = (a + z) - b;
	} else {
		tmp = z + ((a * y) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -1.32e+147:
		tmp = z
	elif x <= 4.2e+183:
		tmp = (a + z) - b
	else:
		tmp = z + ((a * y) / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1.32e+147)
		tmp = z;
	elseif (x <= 4.2e+183)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(z + Float64(Float64(a * y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -1.32e+147)
		tmp = z;
	elseif (x <= 4.2e+183)
		tmp = (a + z) - b;
	else
		tmp = z + ((a * y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1.32e+147], z, If[LessEqual[x, 4.2e+183], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(z + N[(N[(a * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3200000000000001e147

   1. Initial program 61.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 33.6%

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

   if -1.3200000000000001e147 < x < 4.2e183

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 54.9%

         \[\leadsto \left(a + z\right) - b \]

   4. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   if 4.2e183 < x

   1. Initial program 61.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 40.1%

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

   4. Applied rewrites 40.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 34.0%

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

   7. Applied rewrites 34.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 34.0%

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{x} - \frac{b}{x}, y, z\right) \]

      9. sub-div N/A

         \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

      11. lower--.f64 34.1%

          \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

   9. Applied rewrites 34.1%

      \[\leadsto \mathsf{fma}\left(\frac{a - b}{x}, y, z\right) \]

   10. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 30.6%

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

   12. Applied rewrites 30.6%

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

Alternative 12: 58.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+183}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= x -1.32e+147) z (if (<= x 1.05e+183) (- (+ a z) b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.32e+147) {
		tmp = z;
	} else if (x <= 1.05e+183) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	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)
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) :: tmp
    if (x <= (-1.32d+147)) then
        tmp = z
    else if (x <= 1.05d+183) then
        tmp = (a + z) - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.32e+147) {
		tmp = z;
	} else if (x <= 1.05e+183) {
		tmp = (a + z) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -1.32e+147:
		tmp = z
	elif x <= 1.05e+183:
		tmp = (a + z) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1.32e+147)
		tmp = z;
	elseif (x <= 1.05e+183)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -1.32e+147)
		tmp = z;
	elseif (x <= 1.05e+183)
		tmp = (a + z) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1.32e+147], z, If[LessEqual[x, 1.05e+183], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3200000000000001e147 or 1.05e183 < x

   1. Initial program 61.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 33.6%

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

   if -1.3200000000000001e147 < x < 1.05e183

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 54.9%

         \[\leadsto \left(a + z\right) - b \]

   4. Applied rewrites 54.9%

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

Alternative 13: 47.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+71}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= x -2.1e+71) z (if (<= x 4.2e+47) (- a b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.1e+71) {
		tmp = z;
	} else if (x <= 4.2e+47) {
		tmp = a - b;
	} else {
		tmp = z;
	}
	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)
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) :: tmp
    if (x <= (-2.1d+71)) then
        tmp = z
    else if (x <= 4.2d+47) then
        tmp = a - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.1e+71) {
		tmp = z;
	} else if (x <= 4.2e+47) {
		tmp = a - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.1e+71:
		tmp = z
	elif x <= 4.2e+47:
		tmp = a - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.1e+71)
		tmp = z;
	elseif (x <= 4.2e+47)
		tmp = Float64(a - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.1e+71)
		tmp = z;
	elseif (x <= 4.2e+47)
		tmp = a - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.1e+71], z, If[LessEqual[x, 4.2e+47], N[(a - b), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0999999999999999e71 or 4.2e47 < x

   1. Initial program 61.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 33.6%

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

   if -2.0999999999999999e71 < x < 4.2e47

   1. Initial program 61.8%

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

   2. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \left(a + z\right) - \color{blue}{b} \]

      2. lower-+.f64 54.9%

         \[\leadsto \left(a + z\right) - b \]

   4. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   5. Taylor expanded in z around 0

      \[\leadsto a - b \]
   6. Step-by-step derivation

   7. Applied rewrites 35.7%

      \[\leadsto a - b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 44.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00115:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  (if (<= x -0.00115) z (if (<= x 750.0) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -0.00115) {
		tmp = z;
	} else if (x <= 750.0) {
		tmp = a;
	} else {
		tmp = z;
	}
	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)
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) :: tmp
    if (x <= (-0.00115d0)) then
        tmp = z
    else if (x <= 750.0d0) then
        tmp = a
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -0.00115) {
		tmp = z;
	} else if (x <= 750.0) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -0.00115:
		tmp = z
	elif x <= 750.0:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -0.00115)
		tmp = z;
	elseif (x <= 750.0)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -0.00115)
		tmp = z;
	elseif (x <= 750.0)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -0.00115], z, If[LessEqual[x, 750.0], a, z]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00115 or 750 < x

   1. Initial program 61.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 33.6%

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

   if -0.00115 < x < 750

   1. Initial program 61.8%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 31.8%

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

Alternative 15: 31.8% accurate, 29.4× speedup

Math

\[a \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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
    code = a
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return a
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := a

Derivation

1. Initial program 61.8%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 31.8%

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

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64
  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))