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

Percentage Accurate: 74.3% → 90.2%

Time: 11.9s

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: 74.3% 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: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right) \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma b y (fma a t t))) z)
     (if (<= t_1 2e+305)
       t_1
       (if (<= t_1 INFINITY)
         (* (/ y (* (fma (/ b t) y (+ a 1.0)) t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	} else if (t_1 <= 2e+305) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / (fma((b / t), y, (a + 1.0)) * t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	elseif (t_1 <= 2e+305)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / Float64(fma(Float64(b / t), y, Float64(a + 1.0)) * t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + N[(a + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 30.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e305

   1. Initial program 92.2%

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

   if 1.9999999999999999e305 < (/.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 17.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 87.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right) \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right) \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (* (/ y (fma b y (fma a t t))) z)
     (if (<= t_1 2e+305)
       (/ (fma (/ z t) y x) (fma (/ b t) y (+ 1.0 a)))
       (if (<= t_1 INFINITY)
         (* (/ y (* (fma (/ b t) y (+ a 1.0)) t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	} else if (t_1 <= 2e+305) {
		tmp = fma((z / t), y, x) / fma((b / t), y, (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / (fma((b / t), y, (a + 1.0)) * t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	elseif (t_1 <= 2e+305)
		tmp = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / Float64(fma(Float64(b / t), y, Float64(a + 1.0)) * t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + N[(a + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 30.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e305

   1. Initial program 92.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 84.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 85.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if 1.9999999999999999e305 < (/.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 17.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 87.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right) \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right) \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -4e+300)
     (* (/ y (fma b y (fma a t t))) z)
     (if (<= t_1 2e+305)
       (/ x (fma (/ y t) b (+ 1.0 a)))
       (if (<= t_1 INFINITY)
         (* (/ y (* (fma (/ b t) y (+ a 1.0)) t)) z)
         (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -4e+300) {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	} else if (t_1 <= 2e+305) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / (fma((b / t), y, (a + 1.0)) * t)) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= -4e+300)
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	elseif (t_1 <= 2e+305)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / Float64(fma(Float64(b / t), y, Float64(a + 1.0)) * t)) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+300], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + N[(a + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.0000000000000002e300

   1. Initial program 33.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.7%

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

   if -4.0000000000000002e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e305

   1. Initial program 92.2%

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if 1.9999999999999999e305 < (/.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 17.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 87.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{+300}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right) \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (* (/ y (fma b y (fma a t t))) z)))
   (if (<= t_1 -4e+300)
     t_2
     (if (<= t_1 2e+305)
       (/ x (fma (/ y t) b (+ 1.0 a)))
       (if (<= t_1 INFINITY) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = (y / fma(b, y, fma(a, t, t))) * z;
	double tmp;
	if (t_1 <= -4e+300) {
		tmp = t_2;
	} else if (t_1 <= 2e+305) {
		tmp = x / fma((y / t), b, (1.0 + a));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z)
	tmp = 0.0
	if (t_1 <= -4e+300)
		tmp = t_2;
	elseif (t_1 <= 2e+305)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(1.0 + a)));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+300], t$95$2, If[LessEqual[t$95$1, 2e+305], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.0000000000000002e300 or 1.9999999999999999e305 < (/.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 26.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.8%

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

   if -4.0000000000000002e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e305

   1. Initial program 92.2%

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 70.8

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

   5. Applied rewrites 70.8%

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

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

   1. Initial program 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

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{+300}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+300} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (or (<= t_1 -4e+300) (not (<= t_1 2e+305))) (/ z b) (/ x (+ 1.0 a)))))

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 <= -4e+300) || !(t_1 <= 2e+305)) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
    if ((t_1 <= (-4d+300)) .or. (.not. (t_1 <= 2d+305))) 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 t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if ((t_1 <= -4e+300) || !(t_1 <= 2e+305)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if (t_1 <= -4e+300) or not (t_1 <= 2e+305):
		tmp = z / b
	else:
		tmp = x / (1.0 + a)
	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 <= -4e+300) || !(t_1 <= 2e+305))
		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)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if ((t_1 <= -4e+300) || ~((t_1 <= 2e+305)))
		tmp = z / b;
	else
		tmp = x / (1.0 + a);
	end
	tmp_2 = 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[Or[LessEqual[t$95$1, -4e+300], N[Not[LessEqual[t$95$1, 2e+305]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.0000000000000002e300 or 1.9999999999999999e305 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 16.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.0000000000000002e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e305

   1. Initial program 92.2%

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

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

      2. lower-+.f64 58.2

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

   5. Applied rewrites 58.2%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{+300} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+305}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+300}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(b, y, t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -4e+300)
     (* y (/ z (fma b y t)))
     (if (<= t_1 2e+305) (/ x (+ 1.0 a)) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -4e+300) {
		tmp = y * (z / fma(b, y, t));
	} else if (t_1 <= 2e+305) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= -4e+300)
		tmp = Float64(y * Float64(z / fma(b, y, t)));
	elseif (t_1 <= 2e+305)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+300], N[(y * N[(z / N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.0000000000000002e300

   1. Initial program 33.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.7%

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

   if -4.0000000000000002e300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e305

   1. Initial program 92.2%

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

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

      2. lower-+.f64 58.2

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

   5. Applied rewrites 58.2%

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

   if 1.9999999999999999e305 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 7.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{+300}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(b, y, t\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+213}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.7e+58)
     t_1
     (if (<= y 1.85e-40)
       (/ x (+ 1.0 a))
       (if (<= y 9.2e+213) (* (/ y (fma b y (fma a t t))) z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -2.7e+58) {
		tmp = t_1;
	} else if (y <= 1.85e-40) {
		tmp = x / (1.0 + a);
	} else if (y <= 9.2e+213) {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -2.7e+58)
		tmp = t_1;
	elseif (y <= 1.85e-40)
		tmp = Float64(x / Float64(1.0 + a));
	elseif (y <= 9.2e+213)
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -2.7e+58], t$95$1, If[LessEqual[y, 1.85e-40], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+213], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7000000000000001e58 or 9.19999999999999992e213 < y

   1. Initial program 47.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 70.9

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

   7. Applied rewrites 70.9%

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

   if -2.7000000000000001e58 < y < 1.84999999999999999e-40

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 1.84999999999999999e-40 < y < 9.19999999999999992e213

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.6%

      \[\leadsto \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+213}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+58} \lor \neg \left(y \leq 2.1 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.7e+58) (not (<= y 2.1e-40)))
   (/ (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 <= -2.7e+58) || !(y <= 2.1e-40)) {
		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 <= -2.7e+58) || !(y <= 2.1e-40))
		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, -2.7e+58], N[Not[LessEqual[y, 2.1e-40]], $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 < -2.7000000000000001e58 or 2.10000000000000018e-40 < y

   1. Initial program 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 61.5

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

   7. Applied rewrites 61.5%

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

   if -2.7000000000000001e58 < y < 2.10000000000000018e-40

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+58} \lor \neg \left(y \leq 2.1 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.25e+53)
   (/ x a)
   (if (<= a -1.05e-262) (/ z b) (if (<= a 3e-21) (fma (- a) x x) (/ x a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.25e+53) {
		tmp = x / a;
	} else if (a <= -1.05e-262) {
		tmp = z / b;
	} else if (a <= 3e-21) {
		tmp = fma(-a, x, x);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.25e+53)
		tmp = Float64(x / a);
	elseif (a <= -1.05e-262)
		tmp = Float64(z / b);
	elseif (a <= 3e-21)
		tmp = fma(Float64(-a), x, x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.25e+53], N[(x / a), $MachinePrecision], If[LessEqual[a, -1.05e-262], N[(z / b), $MachinePrecision], If[LessEqual[a, 3e-21], N[((-a) * x + x), $MachinePrecision], N[(x / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2500000000000001e53 or 2.99999999999999991e-21 < a

   1. Initial program 75.7%

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.6%

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

   if -1.2500000000000001e53 < a < -1.05e-262

   1. Initial program 72.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if -1.05e-262 < a < 2.99999999999999991e-21

   1. Initial program 79.1%

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

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

      2. lower-+.f64 54.0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 54.0%

      \[\leadsto \mathsf{fma}\left(-a, \color{blue}{x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -1.0) || !(a <= 3e-21)) {
		tmp = x / a;
	} 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 ((a <= (-1.0d0)) .or. (.not. (a <= 3d-21))) then
        tmp = x / a
    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 ((a <= -1.0) || !(a <= 3e-21)) {
		tmp = x / a;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -1.0) || !(a <= 3e-21))
		tmp = Float64(x / a);
	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 ((a <= -1.0) || ~((a <= 3e-21)))
		tmp = x / a;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -1 or 2.99999999999999991e-21 < a

   1. Initial program 76.6%

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 52.6%

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

   if -1 < a < 2.99999999999999991e-21

   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 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 40.8%

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

4. Final simplification 46.9%

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

Alternative 11: 40.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -1 or 2.99999999999999991e-21 < a

   1. Initial program 76.6%

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 52.6%

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

   if -1 < a < 2.99999999999999991e-21

   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 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 40.8%

      \[\leadsto \mathsf{fma}\left(-a, \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

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

Alternative 12: 19.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-a, x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

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

   2. lower-+.f64 46.9

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

5. Applied rewrites 46.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 20.8%

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

9. Final simplification 20.8%

   \[\leadsto \mathsf{fma}\left(-a, x, x\right) \]
10. Add Preprocessing

Developer Target 1: 79.1% 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 2024360 
(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))))