Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 74.8% → 89.9%

Time: 13.1s

Alternatives: 17

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

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)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

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)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, 1 + a\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{y}{t\_2}, \frac{x}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma y (/ b t) (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (fma (/ z t) (/ y t_2) (/ x t_2))
     (if (<= t_1 5e+306)
       t_1
       (if (<= t_1 INFINITY)
         (* (/ y (fma (fma (/ b t) y a) t t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(y, (b / t), (1.0 + a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((z / t), (y / t_2), (x / t_2));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(y, Float64(b / t), Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(z / t), Float64(y / t_2), Float64(x / t_2));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z / t), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.2

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

   7. Applied rewrites 80.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 90.5%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 30.1%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 5.5

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

   7. Applied rewrites 5.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 83.0

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

   10. Applied rewrites 83.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 2: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{y}{t\_2}, \frac{x}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma b (/ y t) (+ 1.0 a))))
   (if (<= t_1 (- INFINITY))
     (fma (/ z t) (/ y t_2) (/ x t_2))
     (if (<= t_1 5e+306)
       t_1
       (if (<= t_1 INFINITY)
         (* (/ y (fma (fma (/ b t) y a) t t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((z / t), (y / t_2), (x / t_2));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(z / t), Float64(y / t_2), Float64(x / t_2));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(z / t), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 60.3%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 90.5%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 30.1%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 5.5

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

   7. Applied rewrites 5.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 83.0

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

   10. Applied rewrites 83.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 3: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (/ (fma (/ z t) y x) (fma (/ b t) y (+ a 1.0)))
     (if (<= t_1 5e+306)
       t_1
       (if (<= t_1 INFINITY)
         (* (/ y (fma (fma (/ b t) y a) t t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (a + 1.0));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(a + 1.0)));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.2

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

   7. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. div-add-rev N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 58.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 90.5%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 30.1%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 5.5

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

   7. Applied rewrites 5.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 83.0

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

   10. Applied rewrites 83.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 4: 87.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(y, \frac{b}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (/ (fma (/ z t) y x) (fma (/ b t) y (+ a 1.0)))
     (if (<= t_2 5e+306)
       (/ t_1 (fma y (/ b t) (+ 1.0 a)))
       (if (<= t_2 INFINITY)
         (* (/ y (fma (fma (/ b t) y a) t t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (a + 1.0));
	} else if (t_2 <= 5e+306) {
		tmp = t_1 / fma(y, (b / t), (1.0 + a));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(a + 1.0)));
	elseif (t_2 <= 5e+306)
		tmp = Float64(t_1 / fma(y, Float64(b / t), Float64(1.0 + a)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+306], N[(t$95$1 / N[(y * N[(b / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0

   1. Initial program 33.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.2

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

   5. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.2

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

   7. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. div-add-rev N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 58.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 90.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 89.3

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

   3. Applied rewrites 89.3%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 30.1%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 5.5

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

   7. Applied rewrites 5.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 83.0

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

   10. Applied rewrites 83.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 5: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 5e+292)
     (/ (fma (/ z t) y x) (fma (/ b t) y (+ a 1.0)))
     (if (<= t_1 INFINITY) (* (/ y (fma (fma (/ b t) y a) t t)) z) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= 5e+292) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (a + 1.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= 5e+292)
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(a + 1.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+292], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999996e292

   1. Initial program 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 83.9

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

   7. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. div-add-rev N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 83.5%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 34.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 8.0

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

   7. Applied rewrites 8.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 78.3

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

   10. Applied rewrites 78.3%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 6: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 5e+292)
     (/ (fma y (/ z t) x) (fma b (/ y t) (+ 1.0 a)))
     (if (<= t_1 INFINITY) (* (/ y (fma (fma (/ b t) y a) t t)) z) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= 5e+292) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / fma(fma((b / t), y, a), t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= 5e+292)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+292], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999996e292

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.5

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. frac-2neg N/A

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

      15. *-commutative N/A

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

      16. frac-2neg N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 83.4

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

   3. Applied rewrites 83.4%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 34.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 8.0

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

   7. Applied rewrites 8.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 78.3

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

   10. Applied rewrites 78.3%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

Alternative 7: 75.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{b}{t}, y, a\right)\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{1 + a}\\ t_4 := \frac{t\_2}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{y}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)} \cdot z\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t\_1, t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ b t) y a))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ 1.0 a)))
        (t_4 (/ t_2 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_4 -2e+287)
     (* (/ (/ y t) (fma b (/ y t) (+ 1.0 a))) z)
     (if (<= t_4 -4e-21)
       t_3
       (if (<= t_4 2e-290)
         (/ (fma (/ z t) y x) t_1)
         (if (<= t_4 5e+306)
           t_3
           (if (<= t_4 INFINITY) (* (/ y (fma t_1 t t)) z) (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b / t), y, a);
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / (1.0 + a);
	double t_4 = t_2 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_4 <= -2e+287) {
		tmp = ((y / t) / fma(b, (y / t), (1.0 + a))) * z;
	} else if (t_4 <= -4e-21) {
		tmp = t_3;
	} else if (t_4 <= 2e-290) {
		tmp = fma((z / t), y, x) / t_1;
	} else if (t_4 <= 5e+306) {
		tmp = t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (y / fma(t_1, t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b / t), y, a)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(1.0 + a))
	t_4 = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_4 <= -2e+287)
		tmp = Float64(Float64(Float64(y / t) / fma(b, Float64(y / t), Float64(1.0 + a))) * z);
	elseif (t_4 <= -4e-21)
		tmp = t_3;
	elseif (t_4 <= 2e-290)
		tmp = Float64(fma(Float64(z / t), y, x) / t_1);
	elseif (t_4 <= 5e+306)
		tmp = t_3;
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(y / fma(t_1, t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+287], N[(N[(N[(y / t), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$4, -4e-21], t$95$3, If[LessEqual[t$95$4, 2e-290], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 5e+306], t$95$3, If[LessEqual[t$95$4, Infinity], N[(N[(y / N[(t$95$1 * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000002e287

   1. Initial program 38.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in x around 0

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

      1. lift-/.f64 67.0

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

   7. Applied rewrites 67.0%

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

   if -2.0000000000000002e287 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999963e-21 or 2.0000000000000001e-290 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 99.4%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 75.7

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 75.7%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   if -3.99999999999999963e-21 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e-290

   1. Initial program 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 78.9

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

   7. Applied rewrites 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. div-add-rev N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 81.0%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 68.8%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 42.2

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 31.6

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

   10. Applied rewrites 31.6%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 30.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 51.8

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

   4. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{b}{t}, y, a\right)\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{1 + a}\\ t_4 := \frac{t\_2}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_5 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{t\_1}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{-21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(t\_1, t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ b t) y a))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ 1.0 a)))
        (t_4 (/ t_2 (+ (+ a 1.0) (/ (* y b) t))))
        (t_5 (/ (fma (/ z t) y x) t_1)))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -4e-21)
       t_3
       (if (<= t_4 2e-290)
         t_5
         (if (<= t_4 5e+306)
           t_3
           (if (<= t_4 INFINITY) (* (/ y (fma t_1 t t)) z) (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b / t), y, a);
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / (1.0 + a);
	double t_4 = t_2 / ((a + 1.0) + ((y * b) / t));
	double t_5 = fma((z / t), y, x) / t_1;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -4e-21) {
		tmp = t_3;
	} else if (t_4 <= 2e-290) {
		tmp = t_5;
	} else if (t_4 <= 5e+306) {
		tmp = t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (y / fma(t_1, t, t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b / t), y, a)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(1.0 + a))
	t_4 = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_5 = Float64(fma(Float64(z / t), y, x) / t_1)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -4e-21)
		tmp = t_3;
	elseif (t_4 <= 2e-290)
		tmp = t_5;
	elseif (t_4 <= 5e+306)
		tmp = t_3;
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(y / fma(t_1, t, t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -4e-21], t$95$3, If[LessEqual[t$95$4, 2e-290], t$95$5, If[LessEqual[t$95$4, 5e+306], t$95$3, If[LessEqual[t$95$4, Infinity], N[(N[(y / N[(t$95$1 * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -3.99999999999999963e-21 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e-290

   1. Initial program 70.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 79.1

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

   7. Applied rewrites 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. div-add-rev N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 77.4%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 65.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999963e-21 or 2.0000000000000001e-290 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 99.4%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 75.9

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 75.9%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 70.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 99.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 14.7

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

   4. Applied rewrites 14.7%

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

Alternative 9: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_4 := \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+287}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-89}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_4 (* (/ y (fma (fma (/ b t) y a) t t)) z)))
   (if (<= t_3 -2e+287)
     t_4
     (if (<= t_3 -5e-89)
       t_2
       (if (<= t_3 2e-290)
         (/ x (fma b (/ y t) (+ 1.0 a)))
         (if (<= t_3 5e+306) t_2 (if (<= t_3 INFINITY) t_4 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (1.0 + a);
	double t_3 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_4 = (y / fma(fma((b / t), y, a), t, t)) * z;
	double tmp;
	if (t_3 <= -2e+287) {
		tmp = t_4;
	} else if (t_3 <= -5e-89) {
		tmp = t_2;
	} else if (t_3 <= 2e-290) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_3 <= 5e+306) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_4 = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z)
	tmp = 0.0
	if (t_3 <= -2e+287)
		tmp = t_4;
	elseif (t_3 <= -5e-89)
		tmp = t_2;
	elseif (t_3 <= 2e-290)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_3 <= 5e+306)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+287], t$95$4, If[LessEqual[t$95$3, -5e-89], t$95$2, If[LessEqual[t$95$3, 2e-290], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+306], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$4, N[(z / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000002e287 or 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 34.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 6.8

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

   7. Applied rewrites 6.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 80.5

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

   10. Applied rewrites 80.5%

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

   if -2.0000000000000002e287 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.99999999999999967e-89 or 2.0000000000000001e-290 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 75.0

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

   if -4.99999999999999967e-89 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e-290

   1. Initial program 73.8%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 64.9

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

   4. Applied rewrites 64.9%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 15.5

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

   4. Applied rewrites 15.5%

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

Alternative 10: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a\right)}\\ \mathbf{if}\;a \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ z t) y x) (fma (/ b t) y a))))
   (if (<= a -2.6e-5)
     t_1
     (if (<= a 0.42) (/ (+ x (/ (* y z) t)) (fma b (/ y t) 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z / t), y, x) / fma((b / t), y, a);
	double tmp;
	if (a <= -2.6e-5) {
		tmp = t_1;
	} else if (a <= 0.42) {
		tmp = (x + ((y * z) / t)) / fma(b, (y / t), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, a))
	tmp = 0.0
	if (a <= -2.6e-5)
		tmp = t_1;
	elseif (a <= 0.42)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / fma(b, Float64(y / t), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.6e-5], t$95$1, If[LessEqual[a, 0.42], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.59999999999999984e-5 or 0.419999999999999984 < a

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. *-commutative N/A

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

      12. frac-2neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. div-add-rev N/A

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

      16. associate-/r* N/A

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

   3. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 77.8

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

   5. Applied rewrites 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 76.6

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

   7. Applied rewrites 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. div-add-rev N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 74.9%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 73.5%

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

   if -2.59999999999999984e-5 < a < 0.419999999999999984

   1. Initial program 75.5%

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

   2. Taylor expanded in a around 0

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + \frac{b \cdot y}{t}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\frac{b \cdot y}{t} + \color{blue}{1}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{b \cdot \frac{y}{t} + 1} \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 74.2

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

   4. Applied rewrites 74.2%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -1.4e+107)
     t_1
     (if (<= y 9.2e-29) (/ (+ x (/ (* y z) t)) (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -1.4e+107) {
		tmp = t_1;
	} else if (y <= 9.2e-29) {
		tmp = (x + ((y * z) / t)) / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -1.4e+107)
		tmp = t_1;
	elseif (y <= 9.2e-29)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -1.4e+107], t$95$1, If[LessEqual[y, 9.2e-29], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999992e107 or 9.19999999999999965e-29 < y

   1. Initial program 54.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 29.0

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 61.8

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

   10. Applied rewrites 61.8%

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

   if -1.39999999999999992e107 < y < 9.19999999999999965e-29

   1. Initial program 91.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 72.7

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{1 + \color{blue}{a}} \]

   4. Applied rewrites 72.7%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma y (/ z t) x) (+ 1.0 a))))
   (if (<= t -6.2e-100)
     t_1
     (if (<= t 2.55e-118) (/ (fma t (/ x y) z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (z / t), x) / (1.0 + a);
	double tmp;
	if (t <= -6.2e-100) {
		tmp = t_1;
	} else if (t <= 2.55e-118) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(y, Float64(z / t), x) / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -6.2e-100)
		tmp = t_1;
	elseif (t <= 2.55e-118)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.2e-100], t$95$1, If[LessEqual[t, 2.55e-118], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.1999999999999997e-100 or 2.54999999999999982e-118 < t

   1. Initial program 80.8%

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

   2. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{1} + a} \]

      3. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + a} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   if -6.1999999999999997e-100 < t < 2.54999999999999982e-118

   1. Initial program 62.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 58.3

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

   4. Applied rewrites 58.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 42.7

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 63.7

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

   10. Applied rewrites 63.7%

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

Alternative 13: 64.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -2e+287)
     (/ (fma y (/ z t) x) a)
     (if (<= t_1 5e+306) (/ x (fma b (/ y t) (+ 1.0 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)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -2e+287) {
		tmp = fma(y, (z / t), x) / a;
	} else if (t_1 <= 5e+306) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= -2e+287)
		tmp = Float64(fma(y, Float64(z / t), x) / a);
	elseif (t_1 <= 5e+306)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+287], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000002e287

   1. Initial program 38.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 29.5

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

   4. Applied rewrites 29.5%

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

   if -2.0000000000000002e287 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 90.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-/l* N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 64.9

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

   4. Applied rewrites 64.9%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 11.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 79.6

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

   4. Applied rewrites 79.6%

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

Alternative 14: 59.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.75 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.75e+45) t_1 (if (<= y 9e-29) (/ x (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -2.75e+45) {
		tmp = t_1;
	} else if (y <= 9e-29) {
		tmp = x / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -2.75e+45)
		tmp = t_1;
	elseif (y <= 9e-29)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -2.75e+45], t$95$1, If[LessEqual[y, 9e-29], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.75e45 or 8.9999999999999996e-29 < y

   1. Initial program 56.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 50.9

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

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 29.2

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 60.5

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

   10. Applied rewrites 60.5%

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

   if -2.75e45 < y < 8.9999999999999996e-29

   1. Initial program 92.8%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 58.4

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 56:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -5.4e-78) t_1 (if (<= t 56.0) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -5.4e-78) {
		tmp = t_1;
	} else if (t <= 56.0) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 + a)
    if (t <= (-5.4d-78)) then
        tmp = t_1
    else if (t <= 56.0d0) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -5.4e-78) {
		tmp = t_1;
	} else if (t <= 56.0) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 + a)
	tmp = 0
	if t <= -5.4e-78:
		tmp = t_1
	elif t <= 56.0:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -5.4e-78)
		tmp = t_1;
	elseif (t <= 56.0)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 + a);
	tmp = 0.0;
	if (t <= -5.4e-78)
		tmp = t_1;
	elseif (t <= 56.0)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.4e-78], t$95$1, If[LessEqual[t, 56.0], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.39999999999999987e-78 or 56 < t

   1. Initial program 81.8%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 57.5

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   4. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   if -5.39999999999999987e-78 < t < 56

   1. Initial program 65.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 54.1

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

   4. Applied rewrites 54.1%

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

Alternative 16: 40.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5.9e-62) (/ z b) (if (<= y 9e-29) x (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5.9e-62) {
		tmp = z / b;
	} else if (y <= 9e-29) {
		tmp = x;
	} else {
		tmp = z / b;
	}
	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 (y <= (-5.9d-62)) then
        tmp = z / b
    else if (y <= 9d-29) then
        tmp = x
    else
        tmp = z / b
    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 (y <= -5.9e-62) {
		tmp = z / b;
	} else if (y <= 9e-29) {
		tmp = x;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5.9e-62:
		tmp = z / b
	elif y <= 9e-29:
		tmp = x
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5.9e-62)
		tmp = Float64(z / b);
	elseif (y <= 9e-29)
		tmp = x;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5.9e-62)
		tmp = z / b;
	elseif (y <= 9e-29)
		tmp = x;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5.9e-62], N[(z / b), $MachinePrecision], If[LessEqual[y, 9e-29], x, N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.9000000000000004e-62 or 8.9999999999999996e-29 < y

   1. Initial program 60.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 47.9

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

   4. Applied rewrites 47.9%

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

   if -5.9000000000000004e-62 < y < 8.9999999999999996e-29

   1. Initial program 94.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + \frac{b \cdot y}{t}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{1} + \frac{b \cdot y}{t}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \frac{b \cdot y}{t}} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\frac{b \cdot y}{t} + \color{blue}{1}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{b \cdot \frac{y}{t} + 1} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1\right)} \]

      9. lower-/.f64 48.0

         \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{\color{blue}{t}}, 1\right)} \]

   4. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto x \]
   6. Step-by-step derivation

   7. Applied rewrites 29.2%

      \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 19.2% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

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
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 74.8%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + \frac{b \cdot y}{t}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + \frac{b \cdot y}{t}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{y \cdot z}{t} + x}{\color{blue}{1} + \frac{b \cdot y}{t}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{y \cdot \frac{z}{t} + x}{1 + \frac{b \cdot y}{t}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\color{blue}{1} + \frac{b \cdot y}{t}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + \frac{b \cdot y}{t}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\frac{b \cdot y}{t} + \color{blue}{1}} \]

   7. associate-/l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{b \cdot \frac{y}{t} + 1} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \color{blue}{\frac{y}{t}}, 1\right)} \]

   9. lower-/.f64 44.8

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{\color{blue}{t}}, 1\right)} \]

4. Applied rewrites 44.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto x \]
6. Step-by-step derivation

7. Applied rewrites 19.2%

   \[\leadsto x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64
  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))