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

Percentage Accurate: 59.6% → 99.2%

Time: 5.1s

Alternatives: 15

Speedup: 2.1×

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: 59.6% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(t + x\right) + y\\ t_3 := \mathsf{fma}\left(z, \frac{y + x}{t\_2}, \left(t + y\right) \cdot \frac{a}{t\_2} - b \cdot \frac{y}{t\_2}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+241}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 (+ (+ t x) y))
        (t_3
         (fma z (/ (+ y x) t_2) (- (* (+ t y) (/ a t_2)) (* b (/ y t_2))))))
   (if (<= t_1 (- INFINITY)) t_3 (if (<= t_1 2e+241) t_1 t_3))))

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 = (t + x) + y;
	double t_3 = fma(z, ((y + x) / t_2), (((t + y) * (a / t_2)) - (b * (y / t_2))));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= 2e+241) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(t + x) + y)
	t_3 = fma(z, Float64(Float64(y + x) / t_2), Float64(Float64(Float64(t + y) * Float64(a / t_2)) - Float64(b * Float64(y / t_2))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= 2e+241)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(N[(t + y), $MachinePrecision] * N[(a / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(b * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, 2e+241], t$95$1, 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 2.0000000000000001e241 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e241

   1. Initial program 59.6%

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

Alternative 2: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(t + x\right) + y\\ t_3 := \mathsf{fma}\left(z, \frac{y + x}{t\_2}, a - b \cdot \frac{y}{t\_2}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+242}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 (+ (+ t x) y))
        (t_3 (fma z (/ (+ y x) t_2) (- a (* b (/ y t_2))))))
   (if (<= t_1 -2e+242) t_3 (if (<= t_1 1e+71) t_1 t_3))))

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 = (t + x) + y;
	double t_3 = fma(z, ((y + x) / t_2), (a - (b * (y / t_2))));
	double tmp;
	if (t_1 <= -2e+242) {
		tmp = t_3;
	} else if (t_1 <= 1e+71) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(t + x) + y)
	t_3 = fma(z, Float64(Float64(y + x) / t_2), Float64(a - Float64(b * Float64(y / t_2))))
	tmp = 0.0
	if (t_1 <= -2e+242)
		tmp = t_3;
	elseif (t_1 <= 1e+71)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(a - N[(b * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+242], t$95$3, If[LessEqual[t$95$1, 1e+71], t$95$1, 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)) < -2.0000000000000001e242 or 1e71 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

   if -2.0000000000000001e242 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e71

   1. Initial program 59.6%

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

Alternative 3: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \left(t + x\right) + y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+242}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y + x}{t\_2}, a - b\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1, a - b \cdot \frac{y}{t\_2}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (+ (+ t x) y)))
   (if (<= t_1 -2e+242)
     (fma z (/ (+ y x) t_2) (- a b))
     (if (<= t_1 1e+135) t_1 (fma z 1.0 (- a (* b (/ y t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = (t + x) + y;
	double tmp;
	if (t_1 <= -2e+242) {
		tmp = fma(z, ((y + x) / t_2), (a - b));
	} else if (t_1 <= 1e+135) {
		tmp = t_1;
	} else {
		tmp = fma(z, 1.0, (a - (b * (y / t_2))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = Float64(Float64(t + x) + y)
	tmp = 0.0
	if (t_1 <= -2e+242)
		tmp = fma(z, Float64(Float64(y + x) / t_2), Float64(a - b));
	elseif (t_1 <= 1e+135)
		tmp = t_1;
	else
		tmp = fma(z, 1.0, Float64(a - Float64(b * Float64(y / t_2))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+242], N[(z * N[(N[(y + x), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+135], t$95$1, N[(z * 1.0 + N[(a - N[(b * N[(y / t$95$2), $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)) < -2.0000000000000001e242

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   if -2.0000000000000001e242 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e134

   1. Initial program 59.6%

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

   if 9.9999999999999996e134 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 70.8%

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

Alternative 4: 81.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left(x + y\right) \cdot z\\ t_2 := \frac{\left(t\_1 + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_3 := \left(t + x\right) + y\\ t_4 := a - \frac{b \cdot y - t\_1}{t\_3}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y + x}{t\_3}, a - b\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+135}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 1, a - b \cdot \frac{y}{t\_3}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ x y) z))
        (t_2 (/ (- (+ t_1 (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_3 (+ (+ t x) y))
        (t_4 (- a (/ (- (* b y) t_1) t_3))))
   (if (<= t_2 (- INFINITY))
     (fma z (/ (+ y x) t_3) (- a b))
     (if (<= t_2 -2e-16)
       t_4
       (if (<= t_2 1e-48)
         (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y)))
         (if (<= t_2 1e+135) t_4 (fma z 1.0 (- a (* b (/ y t_3))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + y) * z;
	double t_2 = ((t_1 + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_3 = (t + x) + y;
	double t_4 = a - (((b * y) - t_1) / t_3);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(z, ((y + x) / t_3), (a - b));
	} else if (t_2 <= -2e-16) {
		tmp = t_4;
	} else if (t_2 <= 1e-48) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else if (t_2 <= 1e+135) {
		tmp = t_4;
	} else {
		tmp = fma(z, 1.0, (a - (b * (y / t_3))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + y) * z)
	t_2 = Float64(Float64(Float64(t_1 + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_3 = Float64(Float64(t + x) + y)
	t_4 = Float64(a - Float64(Float64(Float64(b * y) - t_1) / t_3))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(z, Float64(Float64(y + x) / t_3), Float64(a - b));
	elseif (t_2 <= -2e-16)
		tmp = t_4;
	elseif (t_2 <= 1e-48)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	elseif (t_2 <= 1e+135)
		tmp = t_4;
	else
		tmp = fma(z, 1.0, Float64(a - Float64(b * Float64(y / t_3))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(a - N[(N[(N[(b * y), $MachinePrecision] - t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * N[(N[(y + x), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-16], t$95$4, If[LessEqual[t$95$2, 1e-48], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+135], t$95$4, N[(z * 1.0 + N[(a - N[(b * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2e-16 or 9.9999999999999997e-49 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e134

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

   10. Applied rewrites 59.0%

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

   if -2e-16 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999997e-49

   1. Initial program 59.6%

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

   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 46.8%

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

   4. Applied rewrites 46.8%

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

   if 9.9999999999999996e134 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 70.8%

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

Alternative 5: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ t_2 := \mathsf{fma}\left(z, 1, a - b \cdot \frac{y}{\left(t + x\right) + y}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t + y, z \cdot \left(x + y\right)\right)}{t + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))
        (t_2 (fma z 1.0 (- a (* b (/ y (+ (+ t x) y)))))))
   (if (<= t_1 -5e+148)
     t_2
     (if (<= t_1 4e+55) (/ (fma a (+ t y) (* z (+ x y))) (+ t (+ x y))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double t_2 = fma(z, 1.0, (a - (b * (y / ((t + x) + y)))));
	double tmp;
	if (t_1 <= -5e+148) {
		tmp = t_2;
	} else if (t_1 <= 4e+55) {
		tmp = fma(a, (t + y), (z * (x + y))) / (t + (x + y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	t_2 = fma(z, 1.0, Float64(a - Float64(b * Float64(y / Float64(Float64(t + x) + y)))))
	tmp = 0.0
	if (t_1 <= -5e+148)
		tmp = t_2;
	elseif (t_1 <= 4e+55)
		tmp = Float64(fma(a, Float64(t + y), Float64(z * Float64(x + y))) / Float64(t + Float64(x + y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * 1.0 + N[(a - N[(b * N[(y / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+148], t$95$2, If[LessEqual[t$95$1, 4e+55], N[(N[(a * N[(t + y), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.0000000000000002e148 or 4e55 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 70.8%

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

   if -5.0000000000000002e148 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e55

   1. Initial program 59.6%

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

   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 46.8%

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

   4. Applied rewrites 46.8%

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

Alternative 6: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(z, 1, a - b \cdot \frac{y}{\left(t + x\right) + y}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot \left(\left(a + z\right) - b\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 z 1.0 (- a (* b (/ y (+ (+ t x) y)))))))
   (if (<= t_2 -1e-12)
     t_3
     (if (<= t_2 4e-7)
       (/ (fma a t (* x z)) (+ t x))
       (if (<= t_2 2e+84) (/ (* y (- (+ a z) b)) t_1) 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(z, 1.0, (a - (b * (y / ((t + x) + y)))));
	double tmp;
	if (t_2 <= -1e-12) {
		tmp = t_3;
	} else if (t_2 <= 4e-7) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (t_2 <= 2e+84) {
		tmp = (y * ((a + z) - b)) / t_1;
	} 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(z, 1.0, Float64(a - Float64(b * Float64(y / Float64(Float64(t + x) + y)))))
	tmp = 0.0
	if (t_2 <= -1e-12)
		tmp = t_3;
	elseif (t_2 <= 4e-7)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (t_2 <= 2e+84)
		tmp = Float64(Float64(y * Float64(Float64(a + z) - b)) / t_1);
	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[(z * 1.0 + N[(a - N[(b * N[(y / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-12], t$95$3, If[LessEqual[t$95$2, 4e-7], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+84], N[(N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

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)) < -9.9999999999999998e-13 or 2.0000000000000001e84 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.4%

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 70.8%

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

   if -9.9999999999999998e-13 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999998e-7

   1. Initial program 59.6%

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

   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 40.2%

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

   4. Applied rewrites 40.2%

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

   if 3.9999999999999998e-7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e84

   1. Initial program 59.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.1%

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

   4. Applied rewrites 30.1%

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

Alternative 7: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \mathsf{fma}\left(z, \frac{y + x}{\left(t + x\right) + y}, a - b\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 10^{+79}:\\ \;\;\;\;\frac{y \cdot \left(\left(a + z\right) - b\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 z (/ (+ y x) (+ (+ t x) y)) (- a b))))
   (if (<= t_2 -1e-12)
     t_3
     (if (<= t_2 4e-7)
       (/ (fma a t (* x z)) (+ t x))
       (if (<= t_2 1e+79) (/ (* y (- (+ a z) b)) t_1) 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(z, ((y + x) / ((t + x) + y)), (a - b));
	double tmp;
	if (t_2 <= -1e-12) {
		tmp = t_3;
	} else if (t_2 <= 4e-7) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (t_2 <= 1e+79) {
		tmp = (y * ((a + z) - b)) / t_1;
	} 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(z, Float64(Float64(y + x) / Float64(Float64(t + x) + y)), Float64(a - b))
	tmp = 0.0
	if (t_2 <= -1e-12)
		tmp = t_3;
	elseif (t_2 <= 4e-7)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (t_2 <= 1e+79)
		tmp = Float64(Float64(y * Float64(Float64(a + z) - b)) / t_1);
	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[(z * N[(N[(y + x), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-12], t$95$3, If[LessEqual[t$95$2, 4e-7], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+79], N[(N[(y * N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

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)) < -9.9999999999999998e-13 or 9.9999999999999997e78 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   if -9.9999999999999998e-13 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999998e-7

   1. Initial program 59.6%

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

   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 40.2%

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

   4. Applied rewrites 40.2%

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

   if 3.9999999999999998e-7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999997e78

   1. Initial program 59.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.1%

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

   4. Applied rewrites 30.1%

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

Alternative 8: 65.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -1000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + x}, a - b\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot t\_3}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 (- (+ a z) b)))
   (if (<= t_2 -1000.0)
     (fma z (/ x (+ t x)) (- a b))
     (if (<= t_2 4e-7)
       (/ (fma a t (* x z)) (+ t x))
       (if (<= t_2 2e+84) (/ (* y t_3) t_1) 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 = (a + z) - b;
	double tmp;
	if (t_2 <= -1000.0) {
		tmp = fma(z, (x / (t + x)), (a - b));
	} else if (t_2 <= 4e-7) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (t_2 <= 2e+84) {
		tmp = (y * t_3) / t_1;
	} 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 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= -1000.0)
		tmp = fma(z, Float64(x / Float64(t + x)), Float64(a - b));
	elseif (t_2 <= 4e-7)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (t_2 <= 2e+84)
		tmp = Float64(Float64(y * t_3) / t_1);
	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[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, -1000.0], N[(z * N[(x / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-7], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+84], N[(N[(y * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1e3

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.7%

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

   9. Applied rewrites 56.7%

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

   if -1e3 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999998e-7

   1. Initial program 59.6%

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

   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 40.2%

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

   4. Applied rewrites 40.2%

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

   if 3.9999999999999998e-7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e84

   1. Initial program 59.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.1%

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

   4. Applied rewrites 30.1%

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

   if 2.0000000000000001e84 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

   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 56.8%

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

   4. Applied rewrites 56.8%

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

Alternative 9: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{t\_1}\\ \mathbf{if}\;t\_2 \leq -1000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + x}, a - b\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot \left(z - b\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]

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 -1000.0)
     (fma z (/ x (+ t x)) (- a b))
     (if (<= t_2 4e-7)
       (/ (fma a t (* x z)) (+ t x))
       (if (<= t_2 2e+84) (/ (* y (- z b)) t_1) (- (+ a z) 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 <= -1000.0) {
		tmp = fma(z, (x / (t + x)), (a - b));
	} else if (t_2 <= 4e-7) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else if (t_2 <= 2e+84) {
		tmp = (y * (z - b)) / t_1;
	} else {
		tmp = (a + z) - 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 <= -1000.0)
		tmp = fma(z, Float64(x / Float64(t + x)), Float64(a - b));
	elseif (t_2 <= 4e-7)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	elseif (t_2 <= 2e+84)
		tmp = Float64(Float64(y * Float64(z - b)) / t_1);
	else
		tmp = Float64(Float64(a + z) - 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, -1000.0], N[(z * N[(x / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-7], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+84], N[(N[(y * N[(z - b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1e3

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.7%

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

   9. Applied rewrites 56.7%

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

   if -1e3 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999998e-7

   1. Initial program 59.6%

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

   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 40.2%

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

   4. Applied rewrites 40.2%

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

   if 3.9999999999999998e-7 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e84

   1. Initial program 59.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 30.1%

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 22.5%

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

   7. Applied rewrites 22.5%

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

   if 2.0000000000000001e84 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

   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 56.8%

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

   4. Applied rewrites 56.8%

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

Alternative 10: 64.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + x}, a - b\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))))
   (if (<= t_1 -1000.0)
     (fma z (/ x (+ t x)) (- a b))
     (if (<= t_1 1e-5) (/ (fma a t (* x z)) (+ t x)) (- (+ a z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = fma(z, (x / (t + x)), (a - b));
	} else if (t_1 <= 1e-5) {
		tmp = fma(a, t, (x * z)) / (t + x);
	} else {
		tmp = (a + z) - b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = fma(z, Float64(x / Float64(t + x)), Float64(a - b));
	elseif (t_1 <= 1e-5)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(t + x));
	else
		tmp = Float64(Float64(a + z) - b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], N[(z * N[(x / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(a - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], N[(N[(a + z), $MachinePrecision] - b), $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)) < -1e3

   1. Initial program 59.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. div-add N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

   3. Applied rewrites 71.3%

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

   4. Taylor expanded in y around inf

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

      1. lower--.f64 64.7%

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

   6. Applied rewrites 64.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.7%

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

   9. Applied rewrites 56.7%

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

   if -1e3 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.0000000000000001e-5

   1. Initial program 59.6%

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

   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 40.2%

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

   4. Applied rewrites 40.2%

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

   if 1.0000000000000001e-5 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 59.6%

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

   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 56.8%

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

   4. Applied rewrites 56.8%

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

Alternative 11: 59.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - a}{t}, x, a\right)\\ \mathbf{if}\;t \leq -1.06 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 23.5:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ (- z a) t) x a)))
   (if (<= t -1.06e+155) t_1 (if (<= t 23.5) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((z - a) / t), x, a);
	double tmp;
	if (t <= -1.06e+155) {
		tmp = t_1;
	} else if (t <= 23.5) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(z - a) / t), x, a)
	tmp = 0.0
	if (t <= -1.06e+155)
		tmp = t_1;
	elseif (t <= 23.5)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] * x + a), $MachinePrecision]}, If[LessEqual[t, -1.06e+155], t$95$1, If[LessEqual[t, 23.5], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e155 or 23.5 < t

   1. Initial program 59.6%

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

   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 40.2%

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 29.2%

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

   7. Applied rewrites 29.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 29.2%

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-div N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 29.3%

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

   9. Applied rewrites 29.3%

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

   if -1.06e155 < t < 23.5

   1. Initial program 59.6%

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

   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 56.8%

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

   4. Applied rewrites 56.8%

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

Alternative 12: 59.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{+155}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+162}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.06e+155) a (if (<= t 2.85e+162) (- (+ a z) b) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.06e+155) {
		tmp = a;
	} else if (t <= 2.85e+162) {
		tmp = (a + z) - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 (t <= (-1.06d+155)) then
        tmp = a
    else if (t <= 2.85d+162) then
        tmp = (a + z) - b
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.06e+155) {
		tmp = a;
	} else if (t <= 2.85e+162) {
		tmp = (a + z) - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.06e+155:
		tmp = a
	elif t <= 2.85e+162:
		tmp = (a + z) - b
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.06e+155)
		tmp = a;
	elseif (t <= 2.85e+162)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.06e+155)
		tmp = a;
	elseif (t <= 2.85e+162)
		tmp = (a + z) - b;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.06e+155], a, If[LessEqual[t, 2.85e+162], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], a]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e155 or 2.85e162 < t

   1. Initial program 59.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.7%

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

   if -1.06e155 < t < 2.85e162

   1. Initial program 59.6%

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

   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 56.8%

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

   4. Applied rewrites 56.8%

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

Alternative 13: 48.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 0.0038:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.1e+16) a (if (<= t 0.0038) (- z b) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.1e+16) {
		tmp = a;
	} else if (t <= 0.0038) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 (t <= (-1.1d+16)) then
        tmp = a
    else if (t <= 0.0038d0) then
        tmp = z - b
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.1e+16) {
		tmp = a;
	} else if (t <= 0.0038) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.1e+16:
		tmp = a
	elif t <= 0.0038:
		tmp = z - b
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.1e+16)
		tmp = a;
	elseif (t <= 0.0038)
		tmp = Float64(z - b);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.1e+16)
		tmp = a;
	elseif (t <= 0.0038)
		tmp = z - b;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.1e+16], a, If[LessEqual[t, 0.0038], N[(z - b), $MachinePrecision], a]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1e16 or 0.0038 < t

   1. Initial program 59.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.7%

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

   if -1.1e16 < t < 0.0038

   1. Initial program 59.6%

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

   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 56.8%

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

   4. Applied rewrites 56.8%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 37.6%

         \[\leadsto z - b \]

   7. Applied rewrites 37.6%

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

Alternative 14: 44.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{-41}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -6.1e-41) a (if (<= t 2.7e+34) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -6.1e-41) {
		tmp = a;
	} else if (t <= 2.7e+34) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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 (t <= (-6.1d-41)) then
        tmp = a
    else if (t <= 2.7d+34) then
        tmp = z
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -6.1e-41) {
		tmp = a;
	} else if (t <= 2.7e+34) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -6.1e-41:
		tmp = a
	elif t <= 2.7e+34:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -6.1e-41)
		tmp = a;
	elseif (t <= 2.7e+34)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -6.1e-41)
		tmp = a;
	elseif (t <= 2.7e+34)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -6.1e-41], a, If[LessEqual[t, 2.7e+34], z, a]]

Derivation

1. Split input into 2 regimes

2. if t < -6.0999999999999999e-41 or 2.7e34 < t

   1. Initial program 59.6%

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

   2. Taylor expanded in t around inf

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

   4. Applied rewrites 32.7%

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

   if -6.0999999999999999e-41 < t < 2.7e34

   1. Initial program 59.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.2%

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

Alternative 15: 32.7% 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 59.6%

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

2. Taylor expanded in t around inf

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

4. Applied rewrites 32.7%

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

Reproduce

herbie shell --seed 2025189 
(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)))