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

Percentage Accurate: 74.8% → 90.3%

Time: 5.1s

Alternatives: 12

Speedup: 0.2×

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: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

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))
     (* z (/ y (* t (+ 1.0 (+ a (/ (* b y) t))))))
     (if (<= t_1 -4e-303)
       t_1
       (if (<= t_1 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_1 INFINITY) t_1 (+ (/ z b) (* (/ t b) (/ x y)))))))))

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 = z * (y / (t * (1.0 + (a + ((b * y) / t)))));
	} else if (t_1 <= -4e-303) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (z / b) + ((t / b) * (x / y));
	}
	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(z * Float64(y / Float64(t * Float64(1.0 + Float64(a + Float64(Float64(b * y) / t))))));
	elseif (t_1 <= -4e-303)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	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[(z * N[(y / N[(t * N[(1.0 + N[(a + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-303], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $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 29.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in x around 0

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 85.5

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

   11. Applied rewrites 85.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999972e-303 or 0.0 < (/.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 94.1%

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

   if -3.99999999999999972e-303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 55.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. +-commutative 68.0

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

      2. +-commutative 68.0

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

      3. *-commutative 68.0

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

      4. associate-*r/ 68.0

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

      5. +-commutative 68.0

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

      6. associate-+l+ 68.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 70.8

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

   10. Applied rewrites 70.8%

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

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   8. Applied rewrites 97.7%

      \[\leadsto \frac{z}{b} + \frac{t}{b} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_3 \leq 10^{+38}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{y}{a} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

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)))))
   (if (<= t_3 (- INFINITY))
     (* z (/ y (* t (+ 1.0 (+ a (/ (* b y) t))))))
     (if (<= t_3 -4e-303)
       t_2
       (if (<= t_3 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_3 1e+38)
           (/ x (fma b (/ y t) (+ 1.0 a)))
           (if (<= t_3 2e+283)
             t_2
             (if (<= t_3 INFINITY)
               (* (/ y a) (/ z t))
               (+ (/ z b) (* (/ t b) (/ x y)))))))))))

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 tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = z * (y / (t * (1.0 + (a + ((b * y) / t)))));
	} else if (t_3 <= -4e-303) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_3 <= 1e+38) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else if (t_3 <= 2e+283) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (y / a) * (z / t);
	} else {
		tmp = (z / b) + ((t / b) * (x / y));
	}
	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)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / Float64(t * Float64(1.0 + Float64(a + Float64(Float64(b * y) / t))))));
	elseif (t_3 <= -4e-303)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_3 <= 1e+38)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_3 <= 2e+283)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(y / a) * Float64(z / t));
	else
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	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]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * N[(y / N[(t * N[(1.0 + N[(a + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-303], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 1e+38], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+283], t$95$2, If[LessEqual[t$95$3, Infinity], N[(N[(y / a), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 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 29.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in x around 0

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 85.5

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

   11. Applied rewrites 85.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999972e-303 or 9.99999999999999977e37 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999991e283

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -3.99999999999999972e-303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 55.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. +-commutative 68.0

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

      2. +-commutative 68.0

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

      3. *-commutative 68.0

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

      4. associate-*r/ 68.0

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

      5. +-commutative 68.0

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

      6. associate-+l+ 68.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 70.8

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

   10. Applied rewrites 70.8%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.99999999999999977e37

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      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 62.0

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

   5. Applied rewrites 62.0%

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

   if 1.99999999999999991e283 < (/.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 38.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. 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

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 26.8

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]
   6. 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-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 27.6

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{y}{a} \cdot \frac{z}{t} \]

      4. lift-/.f64 32.2

         \[\leadsto \frac{y}{a} \cdot \frac{z}{t} \]

   10. Applied rewrites 32.2%

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

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   8. Applied rewrites 97.7%

      \[\leadsto \frac{z}{b} + \frac{t}{b} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+38}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{a} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ t_3 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{y}{a} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (fma b (/ y t) a) 1.0)))
        (t_3 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_3 (- INFINITY))
     (* z (/ y (* t (+ 1.0 (+ a (/ (* b y) t))))))
     (if (<= t_3 -4e-303)
       t_2
       (if (<= t_3 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_3 2e+283)
           t_2
           (if (<= t_3 INFINITY)
             (* (/ y a) (/ z t))
             (+ (/ z b) (* (/ t b) (/ x y))))))))))

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 / (fma(b, (y / t), a) + 1.0);
	double t_3 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = z * (y / (t * (1.0 + (a + ((b * y) / t)))));
	} else if (t_3 <= -4e-303) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_3 <= 2e+283) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (y / a) * (z / t);
	} else {
		tmp = (z / b) + ((t / b) * (x / y));
	}
	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(fma(b, Float64(y / t), a) + 1.0))
	t_3 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / Float64(t * Float64(1.0 + Float64(a + Float64(Float64(b * y) / t))))));
	elseif (t_3 <= -4e-303)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_3 <= 2e+283)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(y / a) * Float64(z / t));
	else
		tmp = Float64(Float64(z / b) + Float64(Float64(t / b) * Float64(x / y)));
	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[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + 1.0), $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]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * N[(y / N[(t * N[(1.0 + N[(a + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-303], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 2e+283], t$95$2, If[LessEqual[t$95$3, Infinity], N[(N[(y / a), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / b), $MachinePrecision] + N[(N[(t / b), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $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))) < -inf.0

   1. Initial program 29.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in x around 0

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-+.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 85.5

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

   11. Applied rewrites 85.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999972e-303 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999991e283

   1. Initial program 99.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. 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

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   if -3.99999999999999972e-303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 55.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. +-commutative 68.0

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

      2. +-commutative 68.0

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

      3. *-commutative 68.0

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

      4. associate-*r/ 68.0

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

      5. +-commutative 68.0

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

      6. associate-+l+ 68.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 70.8

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

   10. Applied rewrites 70.8%

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

   if 1.99999999999999991e283 < (/.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 38.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. 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

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 26.8

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{a}} \]
   6. 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-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 27.6

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{y}{a} \cdot \frac{z}{t} \]

      4. lift-/.f64 32.2

         \[\leadsto \frac{y}{a} \cdot \frac{z}{t} \]

   10. Applied rewrites 32.2%

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

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in x around inf

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

      1. times-frac N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.7

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

   8. Applied rewrites 97.7%

      \[\leadsto \frac{z}{b} + \frac{t}{b} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t \cdot \left(1 + \left(a + \frac{b \cdot y}{t}\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{a} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{t}{b} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{t \cdot x}{b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5e+111)
   (/ (fma t (/ x y) z) b)
   (if (<= b 8.6e+29)
     (/ (+ x (/ (* y z) t)) (+ 1.0 a))
     (+ (/ z b) (/ (/ (* t x) b) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e+111) {
		tmp = fma(t, (x / y), z) / b;
	} else if (b <= 8.6e+29) {
		tmp = (x + ((y * z) / t)) / (1.0 + a);
	} else {
		tmp = (z / b) + (((t * x) / b) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5e+111)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (b <= 8.6e+29)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(t * x) / b) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5e+111], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 8.6e+29], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(t * x), $MachinePrecision] / b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999997e111

   1. Initial program 63.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 62.0

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

   8. Applied rewrites 62.0%

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

      1. +-commutative 62.0

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

      2. +-commutative 62.0

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

      3. *-commutative 62.0

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

      4. associate-*r/ 62.0

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

      5. +-commutative 62.0

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

      6. associate-+l+ 62.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 61.8

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

   10. Applied rewrites 61.8%

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

   if -4.9999999999999997e111 < b < 8.6000000000000006e29

   1. Initial program 81.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if 8.6000000000000006e29 < b

   1. Initial program 64.4%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around inf

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 59.0

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

   8. Applied rewrites 59.0%

      \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{t \cdot x}{b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5e+111)
   (/ (fma t (/ x y) z) b)
   (if (<= b 8.6e+29)
     (/ (+ x (/ (* y z) t)) (+ 1.0 a))
     (/ (+ z (/ (* t x) y)) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e+111) {
		tmp = fma(t, (x / y), z) / b;
	} else if (b <= 8.6e+29) {
		tmp = (x + ((y * z) / t)) / (1.0 + a);
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5e+111)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (b <= 8.6e+29)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5e+111], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 8.6e+29], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999997e111

   1. Initial program 63.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 62.0

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

   8. Applied rewrites 62.0%

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

      1. +-commutative 62.0

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

      2. +-commutative 62.0

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

      3. *-commutative 62.0

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

      4. associate-*r/ 62.0

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

      5. +-commutative 62.0

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

      6. associate-+l+ 62.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 61.8

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

   10. Applied rewrites 61.8%

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

   if -4.9999999999999997e111 < b < 8.6000000000000006e29

   1. Initial program 81.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if 8.6000000000000006e29 < b

   1. Initial program 64.4%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 59.6

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

   8. Applied rewrites 59.6%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+21} \lor \neg \left(y \leq 8.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3e+21) (not (<= y 8.2e+27)))
   (/ (fma t (/ x y) z) b)
   (/ x (fma b (/ y t) (+ 1.0 a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e+21) || !(y <= 8.2e+27)) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = x / fma(b, (y / t), (1.0 + a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3e+21) || !(y <= 8.2e+27))
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3e+21], N[Not[LessEqual[y, 8.2e+27]], $MachinePrecision]], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e21 or 8.2000000000000005e27 < y

   1. Initial program 53.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 60.3

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

   8. Applied rewrites 60.3%

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

      1. +-commutative 60.3

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

      2. +-commutative 60.3

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

      3. *-commutative 60.3

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

      4. associate-*r/ 60.3

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

      5. +-commutative 60.3

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

      6. associate-+l+ 60.3

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 64.0

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

   10. Applied rewrites 64.0%

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

   if -3e21 < y < 8.2000000000000005e27

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      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 68.7

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

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+21} \lor \neg \left(y \leq 8.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5e+111)
   (/ (fma t (/ x y) z) b)
   (if (<= b 8.6e+29)
     (/ (fma y (/ z t) x) (+ 1.0 a))
     (/ (+ z (/ (* t x) y)) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5e+111) {
		tmp = fma(t, (x / y), z) / b;
	} else if (b <= 8.6e+29) {
		tmp = fma(y, (z / t), x) / (1.0 + a);
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5e+111)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (b <= 8.6e+29)
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5e+111], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 8.6e+29], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999997e111

   1. Initial program 63.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 62.0

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

   8. Applied rewrites 62.0%

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

      1. +-commutative 62.0

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

      2. +-commutative 62.0

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

      3. *-commutative 62.0

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

      4. associate-*r/ 62.0

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

      5. +-commutative 62.0

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

      6. associate-+l+ 62.0

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 61.8

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

   10. Applied rewrites 61.8%

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

   if -4.9999999999999997e111 < b < 8.6000000000000006e29

   1. Initial program 81.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. 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 68.7

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

   5. Applied rewrites 68.7%

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

   if 8.6000000000000006e29 < b

   1. Initial program 64.4%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 59.6

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

   8. Applied rewrites 59.6%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-66} \lor \neg \left(y \leq 3.8 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.4e-66) (not (<= y 3.8e+27)))
   (/ (fma t (/ x y) z) b)
   (/ x (+ 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.4e-66) || !(y <= 3.8e+27)) {
		tmp = fma(t, (x / y), z) / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.4e-66) || !(y <= 3.8e+27))
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	else
		tmp = Float64(x / Float64(1.0 + a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.4e-66], N[Not[LessEqual[y, 3.8e+27]], $MachinePrecision]], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.39999999999999997e-66 or 3.80000000000000022e27 < y

   1. Initial program 58.3%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{\frac{t \cdot x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      4. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\color{blue}{t \cdot \left(z \cdot \left(1 + a\right)\right)}}{{b}^{2} \cdot y}\right) \]

      5. associate-/r* N/A

         \[\leadsto \frac{z}{b} + \left(\frac{\frac{t \cdot x}{b}}{y} - \frac{\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}}\right) \]

      6. sub-div N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{z}{b} + \frac{\frac{t \cdot x}{b} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{\color{blue}{y}} \]

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 57.3

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

   8. Applied rewrites 57.3%

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

      1. +-commutative 57.3

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

      2. +-commutative 57.3

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

      3. *-commutative 57.3

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

      4. associate-*r/ 57.3

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

      5. +-commutative 57.3

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

      6. associate-+l+ 57.3

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -3.39999999999999997e-66 < y < 3.80000000000000022e27

   1. Initial program 93.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 61.9

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-66} \lor \neg \left(y \leq 3.8 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-285}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+36}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -9.2e-5)
   (/ x a)
   (if (<= a 2.45e-285)
     (/ z b)
     (if (<= a 6e-163) (/ x 1.0) (if (<= a 6e+36) (/ z b) (/ x a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -9.2e-5) {
		tmp = x / a;
	} else if (a <= 2.45e-285) {
		tmp = z / b;
	} else if (a <= 6e-163) {
		tmp = x / 1.0;
	} else if (a <= 6e+36) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-9.2d-5)) then
        tmp = x / a
    else if (a <= 2.45d-285) then
        tmp = z / b
    else if (a <= 6d-163) then
        tmp = x / 1.0d0
    else if (a <= 6d+36) then
        tmp = z / b
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -9.2e-5) {
		tmp = x / a;
	} else if (a <= 2.45e-285) {
		tmp = z / b;
	} else if (a <= 6e-163) {
		tmp = x / 1.0;
	} else if (a <= 6e+36) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -9.2e-5:
		tmp = x / a
	elif a <= 2.45e-285:
		tmp = z / b
	elif a <= 6e-163:
		tmp = x / 1.0
	elif a <= 6e+36:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -9.2e-5)
		tmp = Float64(x / a);
	elseif (a <= 2.45e-285)
		tmp = Float64(z / b);
	elseif (a <= 6e-163)
		tmp = Float64(x / 1.0);
	elseif (a <= 6e+36)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -9.2e-5)
		tmp = x / a;
	elseif (a <= 2.45e-285)
		tmp = z / b;
	elseif (a <= 6e-163)
		tmp = x / 1.0;
	elseif (a <= 6e+36)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -9.2e-5], N[(x / a), $MachinePrecision], If[LessEqual[a, 2.45e-285], N[(z / b), $MachinePrecision], If[LessEqual[a, 6e-163], N[(x / 1.0), $MachinePrecision], If[LessEqual[a, 6e+36], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.20000000000000001e-5 or 6e36 < a

   1. Initial program 73.2%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. 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 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{1} + a} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.6%

      \[\leadsto \frac{x}{\color{blue}{1} + a} \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{x}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 46.8%

       \[\leadsto \frac{x}{a} \]

   if -9.20000000000000001e-5 < a < 2.44999999999999987e-285 or 6.0000000000000005e-163 < a < 6e36

   1. Initial program 76.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 37.7

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

   5. Applied rewrites 37.7%

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

   if 2.44999999999999987e-285 < a < 6.0000000000000005e-163

   1. Initial program 76.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. 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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{1} + a} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.8%

      \[\leadsto \frac{x}{\color{blue}{1} + a} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.8%

       \[\leadsto \frac{x}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+21} \lor \neg \left(y \leq 8.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3e+21) (not (<= y 8.2e+27))) (/ z b) (/ x (+ 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e+21) || !(y <= 8.2e+27)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3d+21)) .or. (.not. (y <= 8.2d+27))) then
        tmp = z / b
    else
        tmp = x / (1.0d0 + a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3e+21) || !(y <= 8.2e+27)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3e+21) or not (y <= 8.2e+27):
		tmp = z / b
	else:
		tmp = x / (1.0 + a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3e+21) || !(y <= 8.2e+27))
		tmp = Float64(z / b);
	else
		tmp = Float64(x / Float64(1.0 + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3e+21) || ~((y <= 8.2e+27)))
		tmp = z / b;
	else
		tmp = x / (1.0 + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3e+21], N[Not[LessEqual[y, 8.2e+27]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e21 or 8.2000000000000005e27 < y

   1. Initial program 53.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if -3e21 < y < 8.2000000000000005e27

   1. Initial program 92.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{1 + a}} \]

      2. lower-+.f64 59.7

         \[\leadsto \frac{x}{1 + \color{blue}{a}} \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+21} \lor \neg \left(y \leq 8.2 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-72} \lor \neg \left(y \leq 2.4 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.95e-72) (not (<= y 2.4e+23))) (/ z b) (/ x 1.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.95e-72) || !(y <= 2.4e+23)) {
		tmp = z / b;
	} else {
		tmp = x / 1.0;
	}
	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 <= (-2.95d-72)) .or. (.not. (y <= 2.4d+23))) then
        tmp = z / b
    else
        tmp = x / 1.0d0
    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 <= -2.95e-72) || !(y <= 2.4e+23)) {
		tmp = z / b;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.95e-72) or not (y <= 2.4e+23):
		tmp = z / b
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.95e-72) || !(y <= 2.4e+23))
		tmp = Float64(z / b);
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.95e-72) || ~((y <= 2.4e+23)))
		tmp = z / b;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.95e-72], N[Not[LessEqual[y, 2.4e+23]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(x / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9499999999999998e-72 or 2.4e23 < y

   1. Initial program 58.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 49.7

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

   5. Applied rewrites 49.7%

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

   if -2.9499999999999998e-72 < y < 2.4e23

   1. Initial program 93.8%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x + \frac{y \cdot z}{t}}{1 + a}} \]
   4. 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 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{1} + a} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.2%

      \[\leadsto \frac{x}{\color{blue}{1} + a} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.8%

       \[\leadsto \frac{x}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-72} \lor \neg \left(y \leq 2.4 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{b} \end{array} \]

FPCore

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

C

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

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 = z / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z / b), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 34.4

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

5. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\frac{z}{b}} \]
6. Add Preprocessing

Developer Target 1: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		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 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) 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 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		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 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		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[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025086 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1707385670788761/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 1518483551868623/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))