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

Percentage Accurate: 75.2% → 90.6%

Time: 12.6s

Alternatives: 15

Speedup: 0.5×

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: 75.2% 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: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ t_3 := \frac{x}{t\_2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{t\_2}, t\_3\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{b} - \frac{\left(1 + a\right) \cdot z}{b \cdot b}}{y}, t, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(a, t, t \cdot \left(1 + \frac{b \cdot y}{t}\right)\right)}, t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma b (/ y t) (+ 1.0 a)))
        (t_3 (/ x t_2)))
   (if (<= t_1 -5e-304)
     (fma (/ y t) (/ z t_2) t_3)
     (if (<= t_1 0.0)
       (fma (/ (- (/ x b) (/ (* (+ 1.0 a) z) (* b b))) y) t (/ z b))
       (if (<= t_1 5e+254)
         t_1
         (if (<= t_1 INFINITY)
           (fma y (/ z (fma a t (* t (+ 1.0 (/ (* b y) t))))) t_3)
           (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(b, (y / t), (1.0 + a));
	double t_3 = x / t_2;
	double tmp;
	if (t_1 <= -5e-304) {
		tmp = fma((y / t), (z / t_2), t_3);
	} else if (t_1 <= 0.0) {
		tmp = fma((((x / b) - (((1.0 + a) * z) / (b * b))) / y), t, (z / b));
	} else if (t_1 <= 5e+254) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(y, (z / fma(a, t, (t * (1.0 + ((b * y) / t))))), t_3);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(b, Float64(y / t), Float64(1.0 + a))
	t_3 = Float64(x / t_2)
	tmp = 0.0
	if (t_1 <= -5e-304)
		tmp = fma(Float64(y / t), Float64(z / t_2), t_3);
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(Float64(Float64(x / b) - Float64(Float64(Float64(1.0 + a) * z) / Float64(b * b))) / y), t, Float64(z / b));
	elseif (t_1 <= 5e+254)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = fma(y, Float64(z / fma(a, t, Float64(t * Float64(1.0 + Float64(Float64(b * y) / t))))), t_3);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-304], N[(N[(y / t), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(N[(1.0 + a), $MachinePrecision] * z), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * t + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+254], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y * N[(z / N[(a * t + N[(t * N[(1.0 + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.99999999999999965e-304

   1. Initial program 93.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-+r+ N/A

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

   4. Applied rewrites 98.9%

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

   if -4.99999999999999965e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 49.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 71.6%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999994e254

   1. Initial program 99.7%

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

   if 4.99999999999999994e254 < (/.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 41.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-+r+ N/A

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

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

   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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{b} - \frac{\left(1 + a\right) \cdot z}{b \cdot b}}{y}, t, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(a, t, t \cdot \left(1 + \frac{b \cdot y}{t}\right)\right)}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t\_2 \cdot t}, \frac{x}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{b} - \frac{\left(1 + a\right) \cdot z}{b \cdot b}}{y}, t, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma b (/ y t) (+ 1.0 a))))
   (if (<= t_1 -5e-304)
     (fma y (/ z (* t_2 t)) (/ x t_2))
     (if (<= t_1 0.0)
       (fma (/ (- (/ x b) (/ (* (+ 1.0 a) z) (* b b))) y) t (/ z b))
       (if (<= t_1 1e+284) t_1 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(b, (y / t), (1.0 + a));
	double tmp;
	if (t_1 <= -5e-304) {
		tmp = fma(y, (z / (t_2 * t)), (x / t_2));
	} else if (t_1 <= 0.0) {
		tmp = fma((((x / b) - (((1.0 + a) * z) / (b * b))) / y), t, (z / b));
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = fma(b, Float64(y / t), Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -5e-304)
		tmp = fma(y, Float64(z / Float64(t_2 * t)), Float64(x / t_2));
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(Float64(Float64(x / b) - Float64(Float64(Float64(1.0 + a) * z) / Float64(b * b))) / y), t, Float64(z / b));
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-304], N[(y * N[(z / N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(x / b), $MachinePrecision] - N[(N[(N[(1.0 + a), $MachinePrecision] * z), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * t + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], t$95$1, 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))) < -4.99999999999999965e-304

   1. Initial program 93.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-+r+ N/A

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

   4. Applied rewrites 96.1%

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

   if -4.99999999999999965e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 49.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 71.6%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000008e284

   1. Initial program 99.7%

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

   if 1.00000000000000008e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 10.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 79.3

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

   5. Applied rewrites 79.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right) \cdot t}, \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{b} - \frac{\left(1 + a\right) \cdot z}{b \cdot b}}{y}, t, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+284}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-300}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+69} \lor \neg \left(t\_1 \leq 10^{+284}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

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 -1e-300)
     x
     (if (or (<= t_1 2e+69) (not (<= t_1 1e+284))) (/ z b) x))))

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 <= -1e-300) {
		tmp = x;
	} else if ((t_1 <= 2e+69) || !(t_1 <= 1e+284)) {
		tmp = z / b;
	} else {
		tmp = x;
	}
	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 <= (-1d-300)) then
        tmp = x
    else if ((t_1 <= 2d+69) .or. (.not. (t_1 <= 1d+284))) then
        tmp = z / b
    else
        tmp = x
    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 <= -1e-300) {
		tmp = x;
	} else if ((t_1 <= 2e+69) || !(t_1 <= 1e+284)) {
		tmp = z / b;
	} else {
		tmp = x;
	}
	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 <= -1e-300:
		tmp = x
	elif (t_1 <= 2e+69) or not (t_1 <= 1e+284):
		tmp = z / b
	else:
		tmp = x
	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 <= -1e-300)
		tmp = x;
	elseif ((t_1 <= 2e+69) || !(t_1 <= 1e+284))
		tmp = Float64(z / b);
	else
		tmp = x;
	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 <= -1e-300)
		tmp = x;
	elseif ((t_1 <= 2e+69) || ~((t_1 <= 1e+284)))
		tmp = z / b;
	else
		tmp = x;
	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[LessEqual[t$95$1, -1e-300], x, If[Or[LessEqual[t$95$1, 2e+69], N[Not[LessEqual[t$95$1, 1e+284]], $MachinePrecision]], N[(z / b), $MachinePrecision], x]]]

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))) < -1.00000000000000003e-300 or 2.0000000000000001e69 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000008e284

   1. Initial program 94.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto x \]

   if -1.00000000000000003e-300 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.0000000000000001e69 or 1.00000000000000008e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 55.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 52.3

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

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+69} \lor \neg \left(\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+284}\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 0.0)
     (/ (fma y (/ z t) x) (fma b (/ y t) (+ 1.0 a)))
     (if (<= t_1 1e+284) t_1 (/ 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 <= 0.0) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), (1.0 + a));
	} else if (t_1 <= 1e+284) {
		tmp = t_1;
	} 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 <= 0.0)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_1 <= 1e+284)
		tmp = t_1;
	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, 0.0], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], t$95$1, 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))) < -0.0

   1. Initial program 81.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.6

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 84.9

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

   4. Applied rewrites 84.9%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.00000000000000008e284

   1. Initial program 99.7%

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

   if 1.00000000000000008e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 10.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 79.3

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

   5. Applied rewrites 79.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+284}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], Infinity], N[(y * N[(z / N[(a * t + N[(t * N[(1.0 + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $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))) < +inf.0

   1. Initial program 82.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-+r+ N/A

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

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

   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

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

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

Alternative 6: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+284}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) 1e+284)
   (/ (fma y (/ z t) x) (fma b (/ y t) (+ 1.0 a)))
   (/ z b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))) <= 1e+284) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), (1.0 + a));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))) <= 1e+284)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), Float64(1.0 + a)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 1e+284], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $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))) < 1.00000000000000008e284

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 86.4

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-+.f64 87.3

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

   4. Applied rewrites 87.3%

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

   if 1.00000000000000008e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 10.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 79.3

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

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+284}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{1 + a} + \frac{y}{t} \cdot \frac{z}{1 + a}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b)))
   (if (<= y -5.6e+52)
     t_1
     (if (<= y 1.55e-150)
       (+ (/ x (+ 1.0 a)) (* (/ y t) (/ z (+ 1.0 a))))
       (if (<= y 1.7e+109) (/ (fma y (/ z t) x) (fma b (/ y t) 1.0)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double tmp;
	if (y <= -5.6e+52) {
		tmp = t_1;
	} else if (y <= 1.55e-150) {
		tmp = (x / (1.0 + a)) + ((y / t) * (z / (1.0 + a)));
	} else if (y <= 1.7e+109) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	tmp = 0.0
	if (y <= -5.6e+52)
		tmp = t_1;
	elseif (y <= 1.55e-150)
		tmp = Float64(Float64(x / Float64(1.0 + a)) + Float64(Float64(y / t) * Float64(z / Float64(1.0 + a))));
	elseif (y <= 1.7e+109)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -5.6e+52], t$95$1, If[LessEqual[y, 1.55e-150], N[(N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * N[(z / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+109], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.6e52 or 1.70000000000000003e109 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 66.6

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

   8. Applied rewrites 66.6%

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

   if -5.6e52 < y < 1.54999999999999999e-150

   1. Initial program 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-add N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-+r+ N/A

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 87.8

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

   7. Applied rewrites 87.8%

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

   if 1.54999999999999999e-150 < y < 1.70000000000000003e109

   1. Initial program 85.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 76.6

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

   5. Applied rewrites 76.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{1 + a} + \frac{y}{t} \cdot \frac{z}{1 + a}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-54}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b)))
   (if (<= y -5.5e+52)
     t_1
     (if (<= y 2.25e-54)
       (/ (+ x (/ (* y z) t)) (+ 1.0 a))
       (if (<= y 1.7e+109) (/ (fma y (/ z t) x) (fma b (/ y t) 1.0)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double tmp;
	if (y <= -5.5e+52) {
		tmp = t_1;
	} else if (y <= 2.25e-54) {
		tmp = (x + ((y * z) / t)) / (1.0 + a);
	} else if (y <= 1.7e+109) {
		tmp = fma(y, (z / t), x) / fma(b, (y / t), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	tmp = 0.0
	if (y <= -5.5e+52)
		tmp = t_1;
	elseif (y <= 2.25e-54)
		tmp = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(1.0 + a));
	elseif (y <= 1.7e+109)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(b, Float64(y / t), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -5.5e+52], t$95$1, If[LessEqual[y, 2.25e-54], N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+109], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.49999999999999996e52 or 1.70000000000000003e109 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 66.6

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

   8. Applied rewrites 66.6%

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

   if -5.49999999999999996e52 < y < 2.2499999999999999e-54

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if 2.2499999999999999e-54 < y < 1.70000000000000003e109

   1. Initial program 79.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-54}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{1 + a}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.1e+52)
   (/ z b)
   (if (<= y 5.6e-35)
     (/ x (+ 1.0 a))
     (if (<= y 1.55e+108) (/ x (fma b (/ y t) 1.0)) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.1e+52) {
		tmp = z / b;
	} else if (y <= 5.6e-35) {
		tmp = x / (1.0 + a);
	} else if (y <= 1.55e+108) {
		tmp = x / fma(b, (y / t), 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.1e+52)
		tmp = Float64(z / b);
	elseif (y <= 5.6e-35)
		tmp = Float64(x / Float64(1.0 + a));
	elseif (y <= 1.55e+108)
		tmp = Float64(x / fma(b, Float64(y / t), 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.1e+52], N[(z / b), $MachinePrecision], If[LessEqual[y, 5.6e-35], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+108], N[(x / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.09999999999999996e52 or 1.5500000000000001e108 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 56.7

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

   5. Applied rewrites 56.7%

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

   if -6.09999999999999996e52 < y < 5.5999999999999999e-35

   1. Initial program 94.6%

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

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

      2. lower-+.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if 5.5999999999999999e-35 < y < 1.5500000000000001e108

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.3%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+108}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-62} \lor \neg \left(t \leq 2 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3.2e-62) (not (<= t 2e-81)))
   (/ (fma y (/ z t) x) (+ 1.0 a))
   (/ (+ z (/ (* t x) y)) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.2e-62) || !(t <= 2e-81)) {
		tmp = fma(y, (z / t), x) / (1.0 + a);
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3.2e-62) || !(t <= 2e-81))
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(1.0 + a));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3.2e-62], N[Not[LessEqual[t, 2e-81]], $MachinePrecision]], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.20000000000000021e-62 or 1.9999999999999999e-81 < t

   1. Initial program 83.9%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -3.20000000000000021e-62 < t < 1.9999999999999999e-81

   1. Initial program 61.6%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 73.2

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

   8. Applied rewrites 73.2%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-62} \lor \neg \left(t \leq 2 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-62} \lor \neg \left(t \leq 3.15 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -3.2e-62) (not (<= t 3.15e-81)))
   (/ x (fma b (/ y t) (+ 1.0 a)))
   (/ (+ z (/ (* t x) y)) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -3.2e-62) || !(t <= 3.15e-81)) {
		tmp = x / fma(b, (y / t), (1.0 + a));
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -3.2e-62) || !(t <= 3.15e-81))
		tmp = Float64(x / fma(b, Float64(y / t), Float64(1.0 + a)));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -3.2e-62], N[Not[LessEqual[t, 3.15e-81]], $MachinePrecision]], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.20000000000000021e-62 or 3.15000000000000011e-81 < t

   1. Initial program 83.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -3.20000000000000021e-62 < t < 3.15000000000000011e-81

   1. Initial program 61.6%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 73.2

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

   8. Applied rewrites 73.2%

      \[\leadsto \frac{z + \frac{t \cdot x}{y}}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-62} \lor \neg \left(t \leq 3.15 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+50} \lor \neg \left(y \leq 3.2 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.35e+50) (not (<= y 3.2e-61)))
   (/ (+ z (/ (* t x) y)) b)
   (/ x (+ 1.0 a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.35e+50) || !(y <= 3.2e-61)) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-3.35d+50)) .or. (.not. (y <= 3.2d-61))) then
        tmp = (z + ((t * x) / y)) / b
    else
        tmp = x / (1.0d0 + a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.35e+50) || !(y <= 3.2e-61)) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.35e+50) or not (y <= 3.2e-61):
		tmp = (z + ((t * x) / y)) / b
	else:
		tmp = x / (1.0 + a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.35e+50) || !(y <= 3.2e-61))
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	else
		tmp = Float64(x / Float64(1.0 + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.35e+50) || ~((y <= 3.2e-61)))
		tmp = (z + ((t * x) / y)) / b;
	else
		tmp = x / (1.0 + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.35e+50], N[Not[LessEqual[y, 3.2e-61]], $MachinePrecision]], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3499999999999999e50 or 3.2000000000000001e-61 < y

   1. Initial program 59.1%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if -3.3499999999999999e50 < y < 3.2000000000000001e-61

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+50} \lor \neg \left(y \leq 3.2 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 46000000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -7e+103)
   x
   (if (<= t 46000000.0) (/ z b) (if (<= t 2e+260) (/ x a) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7e+103) {
		tmp = x;
	} else if (t <= 46000000.0) {
		tmp = z / b;
	} else if (t <= 2e+260) {
		tmp = x / a;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-7d+103)) then
        tmp = x
    else if (t <= 46000000.0d0) then
        tmp = z / b
    else if (t <= 2d+260) then
        tmp = x / a
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7e+103) {
		tmp = x;
	} else if (t <= 46000000.0) {
		tmp = z / b;
	} else if (t <= 2e+260) {
		tmp = x / a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -7e+103:
		tmp = x
	elif t <= 46000000.0:
		tmp = z / b
	elif t <= 2e+260:
		tmp = x / a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -7e+103)
		tmp = x;
	elseif (t <= 46000000.0)
		tmp = Float64(z / b);
	elseif (t <= 2e+260)
		tmp = Float64(x / a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -7e+103)
		tmp = x;
	elseif (t <= 46000000.0)
		tmp = z / b;
	elseif (t <= 2e+260)
		tmp = x / a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -7e+103], x, If[LessEqual[t, 46000000.0], N[(z / b), $MachinePrecision], If[LessEqual[t, 2e+260], N[(x / a), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if t < -7e103 or 2.00000000000000013e260 < t

   1. Initial program 84.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \]
   7. Step-by-step derivation

   8. Applied rewrites 51.9%

      \[\leadsto x \]

   if -7e103 < t < 4.6e7

   1. Initial program 70.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 48.8

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

   5. Applied rewrites 48.8%

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

   if 4.6e7 < t < 2.00000000000000013e260

   1. Initial program 84.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 46.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 46000000:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+52} \lor \neg \left(y \leq 19000000000\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.1e+52) (not (<= y 19000000000.0))) (/ 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 <= -6.1e+52) || !(y <= 19000000000.0)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.1d+52)) .or. (.not. (y <= 19000000000.0d0))) then
        tmp = z / b
    else
        tmp = x / (1.0d0 + a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.1e+52) || !(y <= 19000000000.0)) {
		tmp = z / b;
	} else {
		tmp = x / (1.0 + a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.1e+52) or not (y <= 19000000000.0):
		tmp = z / b
	else:
		tmp = x / (1.0 + a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.1e+52) || !(y <= 19000000000.0))
		tmp = Float64(z / b);
	else
		tmp = Float64(x / Float64(1.0 + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.1e+52) || ~((y <= 19000000000.0)))
		tmp = z / b;
	else
		tmp = x / (1.0 + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.1e+52], N[Not[LessEqual[y, 19000000000.0]], $MachinePrecision]], N[(z / b), $MachinePrecision], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.09999999999999996e52 or 1.9e10 < y

   1. Initial program 54.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -6.09999999999999996e52 < y < 1.9e10

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 64.5

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

   5. Applied rewrites 64.5%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+52} \lor \neg \left(y \leq 19000000000\right):\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 19.4% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 53.9

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

5. Applied rewrites 53.9%

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

6. Taylor expanded in y around 0

   \[\leadsto x \]
7. Step-by-step derivation

8. Applied rewrites 25.8%

   \[\leadsto x \]

9. Final simplification 25.8%

   \[\leadsto x \]
10. Add Preprocessing

Developer Target 1: 78.5% 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 2025045 
(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))))