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

Percentage Accurate: 75.1% → 88.0%

Time: 6.2s

Alternatives: 12

Speedup: 0.8×

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: 75.1% 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: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{z}{t}}{t\_1}, \frac{x}{t\_1}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(y, \frac{\frac{b}{t}}{a - -1}, 1\right)}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ b t) y (- a -1.0))))
   (if (<= t -2.4e-113)
     (fma y (/ (/ z t) t_1) (/ x t_1))
     (if (<= t 1.8e-89)
       (/ (fma x t (* y z)) (fma y b (fma a t t)))
       (/
        (/ (fma (/ z t) y x) (fma y (/ (/ b t) (- a -1.0)) 1.0))
        (- a -1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b / t), y, (a - -1.0));
	double tmp;
	if (t <= -2.4e-113) {
		tmp = fma(y, ((z / t) / t_1), (x / t_1));
	} else if (t <= 1.8e-89) {
		tmp = fma(x, t, (y * z)) / fma(y, b, fma(a, t, t));
	} else {
		tmp = (fma((z / t), y, x) / fma(y, ((b / t) / (a - -1.0)), 1.0)) / (a - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b / t), y, Float64(a - -1.0))
	tmp = 0.0
	if (t <= -2.4e-113)
		tmp = fma(y, Float64(Float64(z / t) / t_1), Float64(x / t_1));
	elseif (t <= 1.8e-89)
		tmp = Float64(fma(x, t, Float64(y * z)) / fma(y, b, fma(a, t, t)));
	else
		tmp = Float64(Float64(fma(Float64(z / t), y, x) / fma(y, Float64(Float64(b / t) / Float64(a - -1.0)), 1.0)) / Float64(a - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b / t), $MachinePrecision] * y + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e-113], N[(y * N[(N[(z / t), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e-89], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(y * N[(N[(b / t), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.40000000000000012e-113

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 76.4%

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

   if -2.40000000000000012e-113 < t < 1.80000000000000003e-89

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   5. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 69.2

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

   7. Applied rewrites 69.2%

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

   if 1.80000000000000003e-89 < t

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

Alternative 2: 87.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(y, \frac{\frac{b}{t}}{a - -1}, 1\right)}}{a - -1}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (/ (fma (/ z t) y x) (fma y (/ (/ b t) (- a -1.0)) 1.0))
          (- a -1.0))))
   (if (<= t -2.4e-113)
     t_1
     (if (<= t 1.8e-89) (/ (fma x t (* y z)) (fma y b (fma a t t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (fma((z / t), y, x) / fma(y, ((b / t) / (a - -1.0)), 1.0)) / (a - -1.0);
	double tmp;
	if (t <= -2.4e-113) {
		tmp = t_1;
	} else if (t <= 1.8e-89) {
		tmp = fma(x, t, (y * z)) / fma(y, b, fma(a, t, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(fma(Float64(z / t), y, x) / fma(y, Float64(Float64(b / t) / Float64(a - -1.0)), 1.0)) / Float64(a - -1.0))
	tmp = 0.0
	if (t <= -2.4e-113)
		tmp = t_1;
	elseif (t <= 1.8e-89)
		tmp = Float64(fma(x, t, Float64(y * z)) / fma(y, b, fma(a, t, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(y * N[(N[(b / t), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e-113], t$95$1, If[LessEqual[t, 1.8e-89], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.40000000000000012e-113 or 1.80000000000000003e-89 < t

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   if -2.40000000000000012e-113 < t < 1.80000000000000003e-89

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   5. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 69.2

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

   7. Applied rewrites 69.2%

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

Alternative 3: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ z t) y x) (fma (/ b t) y (- a -1.0)))))
   (if (<= t -2.4e-113)
     t_1
     (if (<= t 2.6e-45) (/ (fma x t (* y z)) (fma y b (fma a t t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((z / t), y, x) / fma((b / t), y, (a - -1.0));
	double tmp;
	if (t <= -2.4e-113) {
		tmp = t_1;
	} else if (t <= 2.6e-45) {
		tmp = fma(x, t, (y * z)) / fma(y, b, fma(a, t, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(z / t), y, x) / fma(Float64(b / t), y, Float64(a - -1.0)))
	tmp = 0.0
	if (t <= -2.4e-113)
		tmp = t_1;
	elseif (t <= 2.6e-45)
		tmp = Float64(fma(x, t, Float64(y * z)) / fma(y, b, fma(a, t, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(N[(b / t), $MachinePrecision] * y + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e-113], t$95$1, If[LessEqual[t, 2.6e-45], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.40000000000000012e-113 or 2.59999999999999987e-45 < t

   1. Initial program 75.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 74.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 75.0

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

      17. lift-+.f64 N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 75.0

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

   3. Applied rewrites 75.0%

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

   if -2.40000000000000012e-113 < t < 2.59999999999999987e-45

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   5. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 69.2

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

   7. Applied rewrites 69.2%

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

Alternative 4: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{\frac{t}{z}}}{1 + a}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(a, t, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y (/ t z))) (+ 1.0 a))))
   (if (<= t -2.05e+77)
     t_1
     (if (<= t 2.1e+92) (/ (fma x t (* y z)) (fma y b (fma a t t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + (y / (t / z))) / (1.0 + a);
	double tmp;
	if (t <= -2.05e+77) {
		tmp = t_1;
	} else if (t <= 2.1e+92) {
		tmp = fma(x, t, (y * z)) / fma(y, b, fma(a, t, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(y / Float64(t / z))) / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -2.05e+77)
		tmp = t_1;
	elseif (t <= 2.1e+92)
		tmp = Float64(fma(x, t, Float64(y * z)) / fma(y, b, fma(a, t, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.05e+77], t$95$1, If[LessEqual[t, 2.1e+92], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.05e77 or 2.09999999999999986e92 < t

   1. Initial program 75.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 57.3

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

   6. Applied rewrites 57.3%

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

   if -2.05e77 < t < 2.09999999999999986e92

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   5. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 69.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 69.2

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

   7. Applied rewrites 69.2%

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

Alternative 5: 73.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{y \cdot z}{\mathsf{fma}\left(b, y, \left(a - -1\right) \cdot t\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\frac{t\_1}{a - -1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{x + z \cdot \frac{y}{t}}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ (* y z) (fma b y (* (- a -1.0) t)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -2e-286)
       (/ t_1 (- a -1.0))
       (if (<= t_2 4e-269)
         t_3
         (if (<= t_2 INFINITY) (/ (+ x (* z (/ y t))) (+ 1.0 a)) (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = (y * z) / fma(b, y, ((a - -1.0) * t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -2e-286) {
		tmp = t_1 / (a - -1.0);
	} else if (t_2 <= 4e-269) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (x + (z * (y / t))) / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(Float64(y * z) / fma(b, y, Float64(Float64(a - -1.0) * t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -2e-286)
		tmp = Float64(t_1 / Float64(a - -1.0));
	elseif (t_2 <= 4e-269)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(b * y + N[(N[(a - -1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -2e-286], N[(t$95$1 / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-269], t$95$3, If[LessEqual[t$95$2, Infinity], N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -2.0000000000000001e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e-269

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 40.7

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

   8. Applied rewrites 40.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000001e-286

   1. Initial program 75.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 57.0

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

   6. Applied rewrites 57.0%

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

   if 3.9999999999999998e-269 < (/.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 75.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

   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 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

Alternative 6: 70.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x + z \cdot \frac{y}{t}}{1 + a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{z}{t}}{b} \cdot t\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ (+ x (* z (/ y t))) (+ 1.0 a))))
   (if (<= t_1 -2e-286)
     t_2
     (if (<= t_1 4e-269)
       (* (/ (/ z t) b) t)
       (if (<= t_1 INFINITY) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = (x + (z * (y / t))) / (1.0 + a);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

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 t_2 = (x + (z * (y / t))) / (1.0 + a);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	t_2 = (x + (z * (y / t))) / (1.0 + a)
	tmp = 0
	if t_1 <= -2e-286:
		tmp = t_2
	elif t_1 <= 4e-269:
		tmp = ((z / t) / b) * t
	elif t_1 <= math.inf:
		tmp = t_2
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(Float64(x + Float64(z * Float64(y / t))) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = Float64(Float64(Float64(z / t) / b) * t);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	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));
	t_2 = (x + (z * (y / t))) / (1.0 + a);
	tmp = 0.0;
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = ((z / t) / b) * t;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = z / b;
	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]}, Block[{t$95$2 = N[(N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-286], t$95$2, If[LessEqual[t$95$1, 4e-269], N[(N[(N[(z / t), $MachinePrecision] / b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000001e-286 or 3.9999999999999998e-269 < (/.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 75.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

   if -2.0000000000000001e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e-269

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 27.0

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

   6. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.9

         \[\leadsto \frac{\frac{z}{t}}{b} \cdot t \]

   8. Applied rewrites 25.9%

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

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

   1. Initial program 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

Alternative 7: 69.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{t\_1}{a - -1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{z}{t}}{b} \cdot t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+267}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ t_1 (- a -1.0))))
   (if (<= t_2 -2e-286)
     t_3
     (if (<= t_2 4e-269)
       (* (/ (/ z t) b) t)
       (if (<= t_2 2e+267) 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);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = t_1 / (a - -1.0);
	double tmp;
	if (t_2 <= -2e-286) {
		tmp = t_3;
	} else if (t_2 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_2 <= 2e+267) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x + ((y * z) / t)
    t_2 = t_1 / ((a + 1.0d0) + ((y * b) / t))
    t_3 = t_1 / (a - (-1.0d0))
    if (t_2 <= (-2d-286)) then
        tmp = t_3
    else if (t_2 <= 4d-269) then
        tmp = ((z / t) / b) * t
    else if (t_2 <= 2d+267) then
        tmp = t_3
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = t_1 / (a - -1.0);
	double tmp;
	if (t_2 <= -2e-286) {
		tmp = t_3;
	} else if (t_2 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_2 <= 2e+267) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + ((y * z) / t)
	t_2 = t_1 / ((a + 1.0) + ((y * b) / t))
	t_3 = t_1 / (a - -1.0)
	tmp = 0
	if t_2 <= -2e-286:
		tmp = t_3
	elif t_2 <= 4e-269:
		tmp = ((z / t) / b) * t
	elif t_2 <= 2e+267:
		tmp = t_3
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(t_1 / Float64(a - -1.0))
	tmp = 0.0
	if (t_2 <= -2e-286)
		tmp = t_3;
	elseif (t_2 <= 4e-269)
		tmp = Float64(Float64(Float64(z / t) / b) * t);
	elseif (t_2 <= 2e+267)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + ((y * z) / t);
	t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	t_3 = t_1 / (a - -1.0);
	tmp = 0.0;
	if (t_2 <= -2e-286)
		tmp = t_3;
	elseif (t_2 <= 4e-269)
		tmp = ((z / t) / b) * t;
	elseif (t_2 <= 2e+267)
		tmp = t_3;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-286], t$95$3, If[LessEqual[t$95$2, 4e-269], N[(N[(N[(z / t), $MachinePrecision] / b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$2, 2e+267], t$95$3, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000001e-286 or 3.9999999999999998e-269 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e267

   1. Initial program 75.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lift--.f64 57.0

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

   6. Applied rewrites 57.0%

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

   if -2.0000000000000001e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e-269

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 27.0

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

   6. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.9

         \[\leadsto \frac{\frac{z}{t}}{b} \cdot t \]

   8. Applied rewrites 25.9%

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

   if 1.9999999999999999e267 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

Alternative 8: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a - -1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{z}{t}}{b} \cdot t\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ (fma y (/ z t) x) (- a -1.0))))
   (if (<= t_1 -2e-286)
     t_2
     (if (<= t_1 4e-269)
       (* (/ (/ z t) b) t)
       (if (<= t_1 INFINITY) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma(y, (z / t), x) / (a - -1.0);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(fma(y, Float64(z / t), x) / Float64(a - -1.0))
	tmp = 0.0
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = Float64(Float64(Float64(z / t) / b) * t);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-286], t$95$2, If[LessEqual[t$95$1, 4e-269], N[(N[(N[(z / t), $MachinePrecision] / b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000001e-286 or 3.9999999999999998e-269 < (/.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 75.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 57.0

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

   4. Applied rewrites 57.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 57.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if -2.0000000000000001e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e-269

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 27.0

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

   6. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.9

         \[\leadsto \frac{\frac{z}{t}}{b} \cdot t \]

   8. Applied rewrites 25.9%

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

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

   1. Initial program 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

Alternative 9: 55.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{a - -1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{z}{t}}{b} \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+267}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ x (- a -1.0))))
   (if (<= t_1 -2e-286)
     t_2
     (if (<= t_1 4e-269)
       (* (/ (/ z t) b) t)
       (if (<= t_1 2e+267) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (a - -1.0);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_1 <= 2e+267) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
    t_2 = x / (a - (-1.0d0))
    if (t_1 <= (-2d-286)) then
        tmp = t_2
    else if (t_1 <= 4d-269) then
        tmp = ((z / t) / b) * t
    else if (t_1 <= 2d+267) then
        tmp = t_2
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (a - -1.0);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = ((z / t) / b) * t;
	} else if (t_1 <= 2e+267) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	t_2 = x / (a - -1.0)
	tmp = 0
	if t_1 <= -2e-286:
		tmp = t_2
	elif t_1 <= 4e-269:
		tmp = ((z / t) / b) * t
	elif t_1 <= 2e+267:
		tmp = t_2
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(x / Float64(a - -1.0))
	tmp = 0.0
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = Float64(Float64(Float64(z / t) / b) * t);
	elseif (t_1 <= 2e+267)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	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));
	t_2 = x / (a - -1.0);
	tmp = 0.0;
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = ((z / t) / b) * t;
	elseif (t_1 <= 2e+267)
		tmp = t_2;
	else
		tmp = z / b;
	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]}, Block[{t$95$2 = N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-286], t$95$2, If[LessEqual[t$95$1, 4e-269], N[(N[(N[(z / t), $MachinePrecision] / b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+267], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2.0000000000000001e-286 or 3.9999999999999998e-269 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e267

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 43.1

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

   6. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

         \[\leadsto \frac{x}{a - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x}{a - -1} \]

      5. lift--.f64 43.1

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

   8. Applied rewrites 43.1%

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

   if -2.0000000000000001e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e-269

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 61.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 27.0

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

   6. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.9

         \[\leadsto \frac{\frac{z}{t}}{b} \cdot t \]

   8. Applied rewrites 25.9%

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

   if 1.9999999999999999e267 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

Alternative 10: 55.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{a - -1}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+267}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ x (- a -1.0))))
   (if (<= t_1 -2e-286)
     t_2
     (if (<= t_1 4e-269) (/ z b) (if (<= t_1 2e+267) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (a - -1.0);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = z / b;
	} else if (t_1 <= 2e+267) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
    t_2 = x / (a - (-1.0d0))
    if (t_1 <= (-2d-286)) then
        tmp = t_2
    else if (t_1 <= 4d-269) then
        tmp = z / b
    else if (t_1 <= 2d+267) then
        tmp = t_2
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = x / (a - -1.0);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_2;
	} else if (t_1 <= 4e-269) {
		tmp = z / b;
	} else if (t_1 <= 2e+267) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	t_2 = x / (a - -1.0)
	tmp = 0
	if t_1 <= -2e-286:
		tmp = t_2
	elif t_1 <= 4e-269:
		tmp = z / b
	elif t_1 <= 2e+267:
		tmp = t_2
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(x / Float64(a - -1.0))
	tmp = 0.0
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+267)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	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));
	t_2 = x / (a - -1.0);
	tmp = 0.0;
	if (t_1 <= -2e-286)
		tmp = t_2;
	elseif (t_1 <= 4e-269)
		tmp = z / b;
	elseif (t_1 <= 2e+267)
		tmp = t_2;
	else
		tmp = z / b;
	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]}, Block[{t$95$2 = N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-286], t$95$2, If[LessEqual[t$95$1, 4e-269], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+267], t$95$2, 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))) < -2.0000000000000001e-286 or 3.9999999999999998e-269 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.9999999999999999e267

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 43.1

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

   6. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

         \[\leadsto \frac{x}{a - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x}{a - -1} \]

      5. lift--.f64 43.1

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

   8. Applied rewrites 43.1%

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

   if -2.0000000000000001e-286 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999998e-269 or 1.9999999999999999e267 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

Alternative 11: 43.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-88}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.5e-75) (/ z b) (if (<= y 7e-88) (/ x a) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e-75) {
		tmp = z / b;
	} else if (y <= 7e-88) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.5d-75)) then
        tmp = z / b
    else if (y <= 7d-88) then
        tmp = x / a
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e-75) {
		tmp = z / b;
	} else if (y <= 7e-88) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.5e-75:
		tmp = z / b
	elif y <= 7e-88:
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.5e-75)
		tmp = Float64(z / b);
	elseif (y <= 7e-88)
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.5e-75)
		tmp = z / b;
	elseif (y <= 7e-88)
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.5e-75], N[(z / b), $MachinePrecision], If[LessEqual[y, 7e-88], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000002e-75 or 7.0000000000000002e-88 < y

   1. Initial program 75.1%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 33.2

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

   4. Applied rewrites 33.2%

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

   if -6.5000000000000002e-75 < y < 7.0000000000000002e-88

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 43.1

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

   6. Applied rewrites 43.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 26.6

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

   9. Applied rewrites 26.6%

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

Alternative 12: 33.2% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

2. Taylor expanded in y around inf

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

   1. lower-/.f64 33.2

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

4. Applied rewrites 33.2%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64
  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))