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

Percentage Accurate: 61.0% → 99.0%

Time: 6.9s

Alternatives: 14

Speedup: 0.4×

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: 61.0% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(t + x\right) + y\\ t_4 := \frac{t + y}{t\_3}\\ t_5 := \mathsf{fma}\left(a, t\_4, \left(x + y\right) \cdot \frac{z}{t\_1} - b \cdot \frac{y}{t\_1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(a, t\_4, \frac{z \cdot \left(y + x\right) - b \cdot y}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

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))
        (t_3 (+ (+ t x) y))
        (t_4 (/ (+ t y) t_3))
        (t_5 (fma a t_4 (- (* (+ x y) (/ z t_1)) (* b (/ y t_1))))))
   (if (<= t_2 (- INFINITY))
     t_5
     (if (<= t_2 1e+244) (fma a t_4 (/ (- (* z (+ y x)) (* b y)) t_3)) t_5))))

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 t_3 = (t + x) + y;
	double t_4 = (t + y) / t_3;
	double t_5 = fma(a, t_4, (((x + y) * (z / t_1)) - (b * (y / t_1))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_2 <= 1e+244) {
		tmp = fma(a, t_4, (((z * (y + x)) - (b * y)) / t_3));
	} else {
		tmp = t_5;
	}
	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)
	t_3 = Float64(Float64(t + x) + y)
	t_4 = Float64(Float64(t + y) / t_3)
	t_5 = fma(a, t_4, Float64(Float64(Float64(x + y) * Float64(z / t_1)) - Float64(b * Float64(y / t_1))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_2 <= 1e+244)
		tmp = fma(a, t_4, Float64(Float64(Float64(z * Float64(y + x)) - Float64(b * y)) / t_3));
	else
		tmp = t_5;
	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]}, Block[{t$95$3 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t + y), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(a * t$95$4 + N[(N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$5, If[LessEqual[t$95$2, 1e+244], N[(a * t$95$4 + N[(N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]

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)) < -inf.0 or 1.0000000000000001e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e244

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

Alternative 2: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \frac{z}{t\_1}\\ t_4 := \left(t + x\right) + y\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(a, 1, \left(x + y\right) \cdot t\_3 - y \cdot \frac{b}{t\_1}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+244}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + y}{t\_4}, \frac{z \cdot \left(y + x\right) - b \cdot y}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, x + y, \mathsf{fma}\left(-\frac{y}{t\_1}, b, 1 \cdot a\right)\right)\\ \end{array} \]

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))
        (t_3 (/ z t_1))
        (t_4 (+ (+ t x) y)))
   (if (<= t_2 (- INFINITY))
     (fma a 1.0 (- (* (+ x y) t_3) (* y (/ b t_1))))
     (if (<= t_2 1e+244)
       (fma a (/ (+ t y) t_4) (/ (- (* z (+ y x)) (* b y)) t_4))
       (fma t_3 (+ x y) (fma (- (/ y t_1)) b (* 1.0 a)))))))

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 t_3 = z / t_1;
	double t_4 = (t + x) + y;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(a, 1.0, (((x + y) * t_3) - (y * (b / t_1))));
	} else if (t_2 <= 1e+244) {
		tmp = fma(a, ((t + y) / t_4), (((z * (y + x)) - (b * y)) / t_4));
	} else {
		tmp = fma(t_3, (x + y), fma(-(y / t_1), b, (1.0 * a)));
	}
	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)
	t_3 = Float64(z / t_1)
	t_4 = Float64(Float64(t + x) + y)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(a, 1.0, Float64(Float64(Float64(x + y) * t_3) - Float64(y * Float64(b / t_1))));
	elseif (t_2 <= 1e+244)
		tmp = fma(a, Float64(Float64(t + y) / t_4), Float64(Float64(Float64(z * Float64(y + x)) - Float64(b * y)) / t_4));
	else
		tmp = fma(t_3, Float64(x + y), fma(Float64(-Float64(y / t_1)), b, Float64(1.0 * a)));
	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]}, Block[{t$95$3 = N[(z / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(a * 1.0 + N[(N[(N[(x + y), $MachinePrecision] * t$95$3), $MachinePrecision] - N[(y * N[(b / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+244], N[(a * N[(N[(t + y), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(x + y), $MachinePrecision] + N[((-N[(y / t$95$1), $MachinePrecision]) * b + N[(1.0 * a), $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 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 80.6%

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

   10. Applied rewrites 80.6%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e244

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

   if 1.0000000000000001e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 81.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(x + t\right) + y}, x + y, \mathsf{fma}\left(-\frac{y}{\left(x + t\right) + y}, b, 1 \cdot a\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(x + y\right) \cdot \frac{z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(a, 1, t\_3 - y \cdot \frac{b}{t\_1}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+244}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 1, t\_3 - b \cdot \frac{y}{t\_1}\right)\\ \end{array} \]

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))
        (t_3 (* (+ x y) (/ z t_1))))
   (if (<= t_2 (- INFINITY))
     (fma a 1.0 (- t_3 (* y (/ b t_1))))
     (if (<= t_2 1e+244) t_2 (fma a 1.0 (- t_3 (* b (/ y t_1))))))))

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 t_3 = (x + y) * (z / t_1);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(a, 1.0, (t_3 - (y * (b / t_1))));
	} else if (t_2 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = fma(a, 1.0, (t_3 - (b * (y / t_1))));
	}
	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)
	t_3 = Float64(Float64(x + y) * Float64(z / t_1))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(a, 1.0, Float64(t_3 - Float64(y * Float64(b / t_1))));
	elseif (t_2 <= 1e+244)
		tmp = t_2;
	else
		tmp = fma(a, 1.0, Float64(t_3 - Float64(b * Float64(y / t_1))));
	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]}, Block[{t$95$3 = N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(a * 1.0 + N[(t$95$3 - N[(y * N[(b / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+244], t$95$2, N[(a * 1.0 + N[(t$95$3 - N[(b * N[(y / t$95$1), $MachinePrecision]), $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 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 80.6%

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

   10. Applied rewrites 80.6%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e244

   1. Initial program 61.0%

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

   if 1.0000000000000001e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{1}, \left(x + y\right) \cdot \frac{z}{\left(x + t\right) + y} - b \cdot \frac{y}{\left(x + t\right) + y}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \frac{z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(a, 1, \left(x + y\right) \cdot t\_3 - y \cdot \frac{b}{t\_1}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+244}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, x + y, \mathsf{fma}\left(-\frac{y}{t\_1}, b, 1 \cdot a\right)\right)\\ \end{array} \]

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))
        (t_3 (/ z t_1)))
   (if (<= t_2 (- INFINITY))
     (fma a 1.0 (- (* (+ x y) t_3) (* y (/ b t_1))))
     (if (<= t_2 1e+244)
       t_2
       (fma t_3 (+ x y) (fma (- (/ y t_1)) b (* 1.0 a)))))))

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 t_3 = z / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(a, 1.0, (((x + y) * t_3) - (y * (b / t_1))));
	} else if (t_2 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = fma(t_3, (x + y), fma(-(y / t_1), b, (1.0 * a)));
	}
	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)
	t_3 = Float64(z / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(a, 1.0, Float64(Float64(Float64(x + y) * t_3) - Float64(y * Float64(b / t_1))));
	elseif (t_2 <= 1e+244)
		tmp = t_2;
	else
		tmp = fma(t_3, Float64(x + y), fma(Float64(-Float64(y / t_1)), b, Float64(1.0 * a)));
	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]}, Block[{t$95$3 = N[(z / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(a * 1.0 + N[(N[(N[(x + y), $MachinePrecision] * t$95$3), $MachinePrecision] - N[(y * N[(b / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+244], t$95$2, N[(t$95$3 * N[(x + y), $MachinePrecision] + N[((-N[(y / t$95$1), $MachinePrecision]) * b + N[(1.0 * a), $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 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 80.6%

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

   10. Applied rewrites 80.6%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e244

   1. Initial program 61.0%

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

   if 1.0000000000000001e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 81.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(x + t\right) + y}, x + y, \mathsf{fma}\left(-\frac{y}{\left(x + t\right) + y}, b, 1 \cdot a\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(a, 1, \left(x + y\right) \cdot \frac{z}{t\_1} - b \cdot \frac{y}{t\_1}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+244}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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))
        (t_3 (fma a 1.0 (- (* (+ x y) (/ z t_1)) (* b (/ y t_1))))))
   (if (<= t_2 (- INFINITY)) t_3 (if (<= t_2 1e+244) t_2 t_3))))

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 t_3 = fma(a, 1.0, (((x + y) * (z / t_1)) - (b * (y / t_1))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 1e+244) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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)
	t_3 = fma(a, 1.0, Float64(Float64(Float64(x + y) * Float64(z / t_1)) - Float64(b * Float64(y / t_1))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 1e+244)
		tmp = t_2;
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(a * 1.0 + N[(N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 1e+244], t$95$2, t$95$3]]]]]

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)) < -inf.0 or 1.0000000000000001e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e244

   1. Initial program 61.0%

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

Alternative 6: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, \left(x + y\right) \cdot \frac{z}{t\_1} - b\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+271}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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))
        (t_3 (fma a (/ (+ t y) (+ (+ t x) y)) (- (* (+ x y) (/ z t_1)) b))))
   (if (<= t_2 (- INFINITY)) t_3 (if (<= t_2 5e+271) t_2 t_3))))

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 t_3 = fma(a, ((t + y) / ((t + x) + y)), (((x + y) * (z / t_1)) - b));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+271) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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)
	t_3 = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), Float64(Float64(Float64(x + y) * Float64(z / t_1)) - b))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+271)
		tmp = t_2;
	else
		tmp = t_3;
	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]}, Block[{t$95$3 = N[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+271], t$95$2, t$95$3]]]]]

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)) < -inf.0 or 5.0000000000000003e271 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.7%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000003e271

   1. Initial program 61.0%

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

Alternative 7: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z - b\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+271}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 1, \left(x + y\right) \cdot \frac{z}{t\_1} - b\right)\\ \end{array} \]

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))
     (fma a (/ (+ t y) (+ (+ t x) y)) (- z b))
     (if (<= t_2 5e+271) t_2 (fma a 1.0 (- (* (+ x y) (/ z t_1)) 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 = fma(a, ((t + y) / ((t + x) + y)), (z - b));
	} else if (t_2 <= 5e+271) {
		tmp = t_2;
	} else {
		tmp = fma(a, 1.0, (((x + y) * (z / t_1)) - 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 = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), Float64(z - b));
	elseif (t_2 <= 5e+271)
		tmp = t_2;
	else
		tmp = fma(a, 1.0, Float64(Float64(Float64(x + y) * Float64(z / t_1)) - 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[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+271], t$95$2, N[(a * 1.0 + N[(N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] - b), $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 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 63.3%

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

   6. Applied rewrites 63.3%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000003e271

   1. Initial program 61.0%

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

   if 5.0000000000000003e271 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(a, 1, \left(x + y\right) \cdot \frac{z}{\left(x + t\right) + y} - \color{blue}{b}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z - b\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 1, \left(x + y\right) \cdot \frac{z}{t\_1} - b\right)\\ \end{array} \]

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 -2e+137)
     (fma a (/ (+ t y) (+ (+ t x) y)) (- z b))
     (if (<= t_2 1e+244)
       (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y)))
       (fma a 1.0 (- (* (+ x y) (/ z t_1)) 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 <= -2e+137) {
		tmp = fma(a, ((t + y) / ((t + x) + y)), (z - b));
	} else if (t_2 <= 1e+244) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else {
		tmp = fma(a, 1.0, (((x + y) * (z / t_1)) - 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 <= -2e+137)
		tmp = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), Float64(z - b));
	elseif (t_2 <= 1e+244)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	else
		tmp = fma(a, 1.0, Float64(Float64(Float64(x + y) * Float64(z / t_1)) - 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, -2e+137], N[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+244], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * 1.0 + N[(N[(N[(x + y), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision] - b), $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)) < -2.0000000000000001e137

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 63.3%

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

   6. Applied rewrites 63.3%

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

   if -2.0000000000000001e137 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e244

   1. Initial program 61.0%

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

   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 47.9%

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

   4. Applied rewrites 47.9%

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

   if 1.0000000000000001e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 94.3%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 94.3%

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left(a, 1, \left(x + y\right) \cdot \frac{z}{\left(x + t\right) + y} - \color{blue}{b}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \mathsf{fma}\left(a, \frac{t + y}{\left(t + x\right) + y}, z - b\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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 (fma a (/ (+ t y) (+ (+ t x) y)) (- z b))))
   (if (<= t_1 -5e+34)
     t_2
     (if (<= t_1 50000000.0) (/ (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 = fma(a, ((t + y) / ((t + x) + y)), (z - b));
	double tmp;
	if (t_1 <= -5e+34) {
		tmp = t_2;
	} else if (t_1 <= 50000000.0) {
		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 = fma(a, Float64(Float64(t + y) / Float64(Float64(t + x) + y)), Float64(z - b))
	tmp = 0.0
	if (t_1 <= -5e+34)
		tmp = t_2;
	elseif (t_1 <= 50000000.0)
		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[(a * N[(N[(t + y), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+34], t$95$2, If[LessEqual[t$95$1, 50000000.0], 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)) < -4.9999999999999998e34 or 5e7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 72.7%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 63.3%

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

   6. Applied rewrites 63.3%

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

   if -4.9999999999999998e34 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5e7

   1. Initial program 61.0%

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

   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 41.0%

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

   4. Applied rewrites 41.0%

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

Alternative 10: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := z - \left(b - a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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 (- z (- b a))))
   (if (<= t_1 -5e+34)
     t_2
     (if (<= t_1 50000000.0) (/ (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 = z - (b - a);
	double tmp;
	if (t_1 <= -5e+34) {
		tmp = t_2;
	} else if (t_1 <= 50000000.0) {
		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(z - Float64(b - a))
	tmp = 0.0
	if (t_1 <= -5e+34)
		tmp = t_2;
	elseif (t_1 <= 50000000.0)
		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[(z - N[(b - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+34], t$95$2, If[LessEqual[t$95$1, 50000000.0], 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)) < -4.9999999999999998e34 or 5e7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

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

   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 55.2%

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

   4. Applied rewrites 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. sub-negate-rev N/A

         \[\leadsto z + \left(\mathsf{neg}\left(\left(b - a\right)\right)\right) \]

      6. sub-flip-reverse N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 55.2%

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

   6. Applied rewrites 55.2%

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

   if -4.9999999999999998e34 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5e7

   1. Initial program 61.0%

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

   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 41.0%

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

   4. Applied rewrites 41.0%

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

Alternative 11: 56.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := z - \left(b - a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 50000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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 (- z (- b a))))
   (if (<= t_1 -5e+34)
     t_2
     (if (<= t_1 50000000.0) (/ (fma a t (* x z)) t) 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 = z - (b - a);
	double tmp;
	if (t_1 <= -5e+34) {
		tmp = t_2;
	} else if (t_1 <= 50000000.0) {
		tmp = fma(a, t, (x * z)) / t;
	} 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(z - Float64(b - a))
	tmp = 0.0
	if (t_1 <= -5e+34)
		tmp = t_2;
	elseif (t_1 <= 50000000.0)
		tmp = Float64(fma(a, t, Float64(x * z)) / t);
	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[(z - N[(b - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+34], t$95$2, If[LessEqual[t$95$1, 50000000.0], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / t), $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)) < -4.9999999999999998e34 or 5e7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 61.0%

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

   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 55.2%

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

   4. Applied rewrites 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. sub-negate-rev N/A

         \[\leadsto z + \left(\mathsf{neg}\left(\left(b - a\right)\right)\right) \]

      6. sub-flip-reverse N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 55.2%

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

   6. Applied rewrites 55.2%

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

   if -4.9999999999999998e34 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5e7

   1. Initial program 61.0%

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

   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 41.0%

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 24.9%

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

Alternative 12: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+42}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;z - \left(b - a\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= x -5.2e+42) z (- z (- b a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.2e+42) {
		tmp = z;
	} else {
		tmp = z - (b - a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 <= (-5.2d+42)) then
        tmp = z
    else
        tmp = z - (b - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -5.2e+42) {
		tmp = z;
	} else {
		tmp = z - (b - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -5.2e+42:
		tmp = z
	else:
		tmp = z - (b - a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -5.2e+42)
		tmp = z;
	else
		tmp = Float64(z - Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -5.2e+42)
		tmp = z;
	else
		tmp = z - (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -5.2e+42], z, N[(z - N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1999999999999998e42

   1. Initial program 61.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.5%

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

   if -5.1999999999999998e42 < x

   1. Initial program 61.0%

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

   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 55.2%

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

   4. Applied rewrites 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. sub-negate-rev N/A

         \[\leadsto z + \left(\mathsf{neg}\left(\left(b - a\right)\right)\right) \]

      6. sub-flip-reverse N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 55.2%

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

   6. Applied rewrites 55.2%

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

Alternative 13: 45.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+29}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-21}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6.5e+29) z (if (<= z 9.2e-21) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -6.5e+29) {
		tmp = z;
	} else if (z <= 9.2e-21) {
		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 (z <= (-6.5d+29)) then
        tmp = z
    else if (z <= 9.2d-21) 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 (z <= -6.5e+29) {
		tmp = z;
	} else if (z <= 9.2e-21) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -6.5e+29:
		tmp = z
	elif z <= 9.2e-21:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -6.5e+29)
		tmp = z;
	elseif (z <= 9.2e-21)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -6.5e+29)
		tmp = z;
	elseif (z <= 9.2e-21)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6.5e+29], z, If[LessEqual[z, 9.2e-21], a, z]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999997e29 or 9.2e-21 < z

   1. Initial program 61.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.5%

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

   if -6.4999999999999997e29 < z < 9.2e-21

   1. Initial program 61.0%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 33.2%

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

Alternative 14: 33.2% accurate, 29.5× 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 61.0%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 33.2%

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

Reproduce

herbie shell --seed 2025195 
(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)))