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

Percentage Accurate: 75.1% → 89.7%

Time: 6.4s

Alternatives: 17

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: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, \frac{z}{t\_1}, \frac{x}{t\_1}\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(y, \frac{\frac{b}{t}}{a - -1}, 1\right)}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ b t) y (- a -1.0))))
   (if (<= t -7.5e-10)
     (fma (/ y t) (/ z t_1) (/ x t_1))
     (if (<= t 1.1e-93)
       (* (fma x t (* y z)) (/ 1.0 (fma b y (fma t a 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 <= -7.5e-10) {
		tmp = fma((y / t), (z / t_1), (x / t_1));
	} else if (t <= 1.1e-93) {
		tmp = fma(x, t, (y * z)) * (1.0 / fma(b, y, fma(t, a, 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 <= -7.5e-10)
		tmp = fma(Float64(y / t), Float64(z / t_1), Float64(x / t_1));
	elseif (t <= 1.1e-93)
		tmp = Float64(fma(x, t, Float64(y * z)) * Float64(1.0 / fma(b, y, fma(t, a, 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, -7.5e-10], N[(N[(y / t), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e-93], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * y + N[(t * a + t), $MachinePrecision]), $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 < -7.49999999999999995e-10

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

      1. lift-fma.f64 N/A

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

   5. Applied rewrites 77.5%

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

   if -7.49999999999999995e-10 < t < 1.09999999999999998e-93

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   5. Applied rewrites 69.5%

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

   if 1.09999999999999998e-93 < 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: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{z}{t}}{t\_1}, \frac{x}{t\_1}\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(y, \frac{\frac{b}{t}}{a - -1}, 1\right)}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ b t) y (- a -1.0))))
   (if (<= t -3.4e-16)
     (fma y (/ (/ z t) t_1) (/ x t_1))
     (if (<= t 1.1e-93)
       (* (fma x t (* y z)) (/ 1.0 (fma b y (fma t a 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 <= -3.4e-16) {
		tmp = fma(y, ((z / t) / t_1), (x / t_1));
	} else if (t <= 1.1e-93) {
		tmp = fma(x, t, (y * z)) * (1.0 / fma(b, y, fma(t, a, 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 <= -3.4e-16)
		tmp = fma(y, Float64(Float64(z / t) / t_1), Float64(x / t_1));
	elseif (t <= 1.1e-93)
		tmp = Float64(fma(x, t, Float64(y * z)) * Float64(1.0 / fma(b, y, fma(t, a, 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, -3.4e-16], N[(y * N[(N[(z / t), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e-93], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * y + N[(t * a + t), $MachinePrecision]), $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 < -3.4e-16

   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 77.0%

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

   if -3.4e-16 < t < 1.09999999999999998e-93

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   5. Applied rewrites 69.5%

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

   if 1.09999999999999998e-93 < 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 3: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\mathsf{fma}\left(y, \frac{\frac{b}{t}}{a - -1}, 1\right)}}{a - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -2e-15)
     (/ t_1 (fma (/ b t) y (- a -1.0)))
     (if (<= t 1.1e-93)
       (* (fma x t (* y z)) (/ 1.0 (fma b y (fma t a t))))
       (/ (/ t_1 (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((z / t), y, x);
	double tmp;
	if (t <= -2e-15) {
		tmp = t_1 / fma((b / t), y, (a - -1.0));
	} else if (t <= 1.1e-93) {
		tmp = fma(x, t, (y * z)) * (1.0 / fma(b, y, fma(t, a, t)));
	} else {
		tmp = (t_1 / 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(z / t), y, x)
	tmp = 0.0
	if (t <= -2e-15)
		tmp = Float64(t_1 / fma(Float64(b / t), y, Float64(a - -1.0)));
	elseif (t <= 1.1e-93)
		tmp = Float64(fma(x, t, Float64(y * z)) * Float64(1.0 / fma(b, y, fma(t, a, t))));
	else
		tmp = Float64(Float64(t_1 / 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[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2e-15], N[(t$95$1 / N[(N[(b / t), $MachinePrecision] * y + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e-93], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * y + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / 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 < -2.0000000000000002e-15

   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 75.1

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

      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 75.1

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

   3. Applied rewrites 75.1%

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

   if -2.0000000000000002e-15 < t < 1.09999999999999998e-93

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   5. Applied rewrites 69.5%

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

   if 1.09999999999999998e-93 < 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 4: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot z\right) \cdot \frac{1}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e-15)
     t_1
     (if (<= t 1.1e-93)
       (* (fma x t (* y z)) (/ 1.0 (fma b y (fma t a 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 <= -2e-15) {
		tmp = t_1;
	} else if (t <= 1.1e-93) {
		tmp = fma(x, t, (y * z)) * (1.0 / fma(b, y, fma(t, a, 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 <= -2e-15)
		tmp = t_1;
	elseif (t <= 1.1e-93)
		tmp = Float64(fma(x, t, Float64(y * z)) * Float64(1.0 / fma(b, y, fma(t, a, 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, -2e-15], t$95$1, If[LessEqual[t, 1.1e-93], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b * y + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.0000000000000002e-15 or 1.09999999999999998e-93 < 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 75.1

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

      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 75.1

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

   3. Applied rewrites 75.1%

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

   if -2.0000000000000002e-15 < t < 1.09999999999999998e-93

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   5. Applied rewrites 69.5%

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

Alternative 5: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\frac{t \cdot \left(y \cdot z\right)}{\mathsf{fma}\left(b, y, t \cdot \left(1 + a\right)\right)}}{t}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (/ (/ (* t (* y z)) (fma b y (* t (+ 1.0 a)))) t)
     (if (<= t_2 2e+279)
       (/ t_1 (fma b (/ y t) (- a -1.0)))
       (/ (+ z (/ (* t x) y)) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((t * (y * z)) / fma(b, y, (t * (1.0 + a)))) / t;
	} else if (t_2 <= 2e+279) {
		tmp = t_1 / fma(b, (y / t), (a - -1.0));
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t * Float64(y * z)) / fma(b, y, Float64(t * Float64(1.0 + a)))) / t);
	elseif (t_2 <= 2e+279)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(a - -1.0)));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(t * N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(b * y + N[(t * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$2, 2e+279], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / 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))) < -inf.0

   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. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 33.7

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

   6. Applied rewrites 33.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000012e279

   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

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 74.7

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

      9. lift-+.f64 N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval 74.7

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

   3. Applied rewrites 74.7%

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

   if 2.00000000000000012e279 < (/.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. 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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

Alternative 6: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(\frac{b}{t}, y, a - -1\right)}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, y \cdot z\right)}{t + b \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -4e-66)
     t_1
     (if (<= t 9e-96) (/ (fma t x (* y z)) (+ t (* b y))) 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 <= -4e-66) {
		tmp = t_1;
	} else if (t <= 9e-96) {
		tmp = fma(t, x, (y * z)) / (t + (b * y));
	} 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 <= -4e-66)
		tmp = t_1;
	elseif (t <= 9e-96)
		tmp = Float64(fma(t, x, Float64(y * z)) / Float64(t + Float64(b * y)));
	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, -4e-66], t$95$1, If[LessEqual[t, 9e-96], N[(N[(t * x + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t + N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9999999999999999e-66 or 9e-96 < 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 75.1

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

      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 75.1

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

   3. Applied rewrites 75.1%

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

   if -3.9999999999999999e-66 < t < 9e-96

   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. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 48.6

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

   6. Applied rewrites 48.6%

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

Alternative 7: 73.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{1 + a}\\ t_4 := \frac{t\_2}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(b, y, t \cdot \left(1 + a\right)\right)}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ 1.0 a)))
        (t_4 (/ t_2 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_4 (- INFINITY))
     (/ (* y z) (fma b y (* t (+ 1.0 a))))
     (if (<= t_4 -2e-304)
       t_3
       (if (<= t_4 0.0) t_1 (if (<= t_4 2e+279) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / (1.0 + a);
	double t_4 = t_2 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (y * z) / fma(b, y, (t * (1.0 + a)));
	} else if (t_4 <= -2e-304) {
		tmp = t_3;
	} else if (t_4 <= 0.0) {
		tmp = t_1;
	} else if (t_4 <= 2e+279) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(1.0 + a))
	t_4 = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(y * z) / fma(b, y, Float64(t * Float64(1.0 + a))));
	elseif (t_4 <= -2e-304)
		tmp = t_3;
	elseif (t_4 <= 0.0)
		tmp = t_1;
	elseif (t_4 <= 2e+279)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(y * z), $MachinePrecision] / N[(b * y + N[(t * N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e-304], t$95$3, If[LessEqual[t$95$4, 0.0], t$95$1, If[LessEqual[t$95$4, 2e+279], t$95$3, t$95$1]]]]]]]]

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))) < -inf.0

   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. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 41.7

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

   6. Applied rewrites 41.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999994e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000012e279

   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 56.8

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

   4. Applied rewrites 56.8%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2.00000000000000012e279 < (/.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. 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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

Alternative 8: 72.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z + \frac{t \cdot x}{y}}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_4 := \frac{t\_2}{1 + a}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, y \cdot z\right)}{a}}{t}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ z (/ (* t x) y)) b))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ (+ a 1.0) (/ (* y b) t))))
        (t_4 (/ t_2 (+ 1.0 a))))
   (if (<= t_3 (- INFINITY))
     (/ (/ (fma t x (* y z)) a) t)
     (if (<= t_3 -2e-304)
       t_4
       (if (<= t_3 0.0) t_1 (if (<= t_3 2e+279) t_4 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + ((t * x) / y)) / b;
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / ((a + 1.0) + ((y * b) / t));
	double t_4 = t_2 / (1.0 + a);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (fma(t, x, (y * z)) / a) / t;
	} else if (t_3 <= -2e-304) {
		tmp = t_4;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 2e+279) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + Float64(Float64(t * x) / y)) / b)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_4 = Float64(t_2 / Float64(1.0 + a))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(fma(t, x, Float64(y * z)) / a) / t);
	elseif (t_3 <= -2e-304)
		tmp = t_4;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 2e+279)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(t * x + N[(y * z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$3, -2e-304], t$95$4, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 2e+279], t$95$4, t$95$1]]]]]]]]

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))) < -inf.0

   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. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 60.9%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 28.7

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

   6. Applied rewrites 28.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999994e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2.00000000000000012e279

   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 56.8

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

   4. Applied rewrites 56.8%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 2.00000000000000012e279 < (/.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. 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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

Alternative 9: 67.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{1 + a}\\ \mathbf{if}\;t \leq -140000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ z t) y x) (+ 1.0 a))))
   (if (<= t -140000.0)
     t_1
     (if (<= t 8.1e+74) (/ (+ z (/ (* t x) y)) b) 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) / (1.0 + a);
	double tmp;
	if (t <= -140000.0) {
		tmp = t_1;
	} else if (t <= 8.1e+74) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(z / t), y, x) / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -140000.0)
		tmp = t_1;
	elseif (t <= 8.1e+74)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	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[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -140000.0], t$95$1, If[LessEqual[t, 8.1e+74], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.4e5 or 8.1000000000000003e74 < 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 75.1

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

      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 75.1

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

   3. Applied rewrites 75.1%

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

   4. Taylor expanded in y around 0

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

      1. lower-+.f64 56.7

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

   6. Applied rewrites 56.7%

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

   if -1.4e5 < t < 8.1000000000000003e74

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

Alternative 10: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -800000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -800000.0)
   (/ (fma x (/ t y) z) b)
   (if (<= y 5.8e+87)
     (/ x (fma b (/ y t) (- a -1.0)))
     (/ (+ z (/ (* t x) y)) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -800000.0) {
		tmp = fma(x, (t / y), z) / b;
	} else if (y <= 5.8e+87) {
		tmp = x / fma(b, (y / t), (a - -1.0));
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -800000.0)
		tmp = Float64(fma(x, Float64(t / y), z) / b);
	elseif (y <= 5.8e+87)
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a - -1.0)));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -800000.0], N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 5.8e+87], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8e5

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.1

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

   8. Applied rewrites 41.1%

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

   if -8e5 < y < 5.7999999999999996e87

   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 x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.7

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

   4. Applied rewrites 51.7%

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 53.2

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

   8. Applied rewrites 53.2%

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

   if 5.7999999999999996e87 < y

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

Alternative 11: 60.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -500000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -500000.0)
   (/ (fma x (/ t y) z) b)
   (if (<= y 2.3e+26) (/ x (- a -1.0)) (/ (+ z (/ (* t x) y)) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -500000.0) {
		tmp = fma(x, (t / y), z) / b;
	} else if (y <= 2.3e+26) {
		tmp = x / (a - -1.0);
	} else {
		tmp = (z + ((t * x) / y)) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -500000.0)
		tmp = Float64(fma(x, Float64(t / y), z) / b);
	elseif (y <= 2.3e+26)
		tmp = Float64(x / Float64(a - -1.0));
	else
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -500000.0], N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 2.3e+26], N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5e5

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.1

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

   8. Applied rewrites 41.1%

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

   if -5e5 < y < 2.3000000000000001e26

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

   if 2.3000000000000001e26 < y

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

Alternative 12: 60.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(x, \frac{t}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma x (/ t y) z) b)))
   (if (<= y -500000.0) t_1 (if (<= y 2.3e+26) (/ x (- a -1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, (t / y), z) / b;
	double tmp;
	if (y <= -500000.0) {
		tmp = t_1;
	} else if (y <= 2.3e+26) {
		tmp = x / (a - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(x, Float64(t / y), z) / b)
	tmp = 0.0
	if (y <= -500000.0)
		tmp = t_1;
	elseif (y <= 2.3e+26)
		tmp = Float64(x / Float64(a - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[(t / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -500000.0], t$95$1, If[LessEqual[y, 2.3e+26], N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e5 or 2.3000000000000001e26 < y

   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. mult-flip N/A

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

      7. lift-+.f64 N/A

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

      8. sum-to-mult N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 77.1%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 41.1

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

   8. Applied rewrites 41.1%

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

   if -5e5 < y < 2.3000000000000001e26

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

Alternative 13: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{a - -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t \cdot \frac{z}{t}}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (/ z b)
     (if (<= t_1 -2e-304)
       t_2
       (if (<= t_1 0.0)
         (/ (* t (/ z t)) b)
         (if (<= t_1 5e+264) 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 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -2e-304) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (t * (z / t)) / b;
	} else if (t_1 <= 5e+264) {
		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 / (a - -1.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z / b;
	} else if (t_1 <= -2e-304) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (t * (z / t)) / b;
	} else if (t_1 <= 5e+264) {
		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 <= -math.inf:
		tmp = z / b
	elif t_1 <= -2e-304:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (t * (z / t)) / b
	elif t_1 <= 5e+264:
		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 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -2e-304)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(t * Float64(z / t)) / b);
	elseif (t_1 <= 5e+264)
		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 <= -Inf)
		tmp = z / b;
	elseif (t_1 <= -2e-304)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (t * (z / t)) / b;
	elseif (t_1 <= 5e+264)
		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, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -2e-304], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(z / t), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 5e+264], 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))) < -inf.0 or 5.00000000000000033e264 < (/.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 33.5

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

   4. Applied rewrites 33.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999994e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.00000000000000033e264

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

   if -1.99999999999999994e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 75.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 25.3

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

   4. Applied rewrites 25.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 26.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 28.2

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

   6. Applied rewrites 28.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 28.2

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

   9. Applied rewrites 28.2%

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

Alternative 14: 58.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{x}{a - -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (/ z b)
     (if (<= t_1 -2e-304)
       t_2
       (if (<= t_1 0.0) (/ z b) (if (<= t_1 5e+264) 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 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -2e-304) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 5e+264) {
		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 / (a - -1.0);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z / b;
	} else if (t_1 <= -2e-304) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 5e+264) {
		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 <= -math.inf:
		tmp = z / b
	elif t_1 <= -2e-304:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = z / b
	elif t_1 <= 5e+264:
		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 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -2e-304)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 5e+264)
		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 <= -Inf)
		tmp = z / b;
	elseif (t_1 <= -2e-304)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = z / b;
	elseif (t_1 <= 5e+264)
		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, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -2e-304], t$95$2, If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 5e+264], 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))) < -inf.0 or -1.99999999999999994e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 5.00000000000000033e264 < (/.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 33.5

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

   4. Applied rewrites 33.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999994e-304 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 5.00000000000000033e264

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

Alternative 15: 44.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-278}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{-112}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;t\_1 \leq 10^{+112}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+253}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 (- INFINITY))
     (/ z b)
     (if (<= t_1 -1e+41)
       (/ x 1.0)
       (if (<= t_1 -2e-304)
         (/ x a)
         (if (<= t_1 4e-278)
           (/ z b)
           (if (<= t_1 1e-112)
             (/ x 1.0)
             (if (<= t_1 1e+112)
               (/ x a)
               (if (<= t_1 1e+253) (/ x 1.0) (/ z b))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z / b;
	} else if (t_1 <= -1e+41) {
		tmp = x / 1.0;
	} else if (t_1 <= -2e-304) {
		tmp = x / a;
	} else if (t_1 <= 4e-278) {
		tmp = z / b;
	} else if (t_1 <= 1e-112) {
		tmp = x / 1.0;
	} else if (t_1 <= 1e+112) {
		tmp = x / a;
	} else if (t_1 <= 1e+253) {
		tmp = x / 1.0;
	} 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 tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z / b;
	} else if (t_1 <= -1e+41) {
		tmp = x / 1.0;
	} else if (t_1 <= -2e-304) {
		tmp = x / a;
	} else if (t_1 <= 4e-278) {
		tmp = z / b;
	} else if (t_1 <= 1e-112) {
		tmp = x / 1.0;
	} else if (t_1 <= 1e+112) {
		tmp = x / a;
	} else if (t_1 <= 1e+253) {
		tmp = x / 1.0;
	} 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))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z / b
	elif t_1 <= -1e+41:
		tmp = x / 1.0
	elif t_1 <= -2e-304:
		tmp = x / a
	elif t_1 <= 4e-278:
		tmp = z / b
	elif t_1 <= 1e-112:
		tmp = x / 1.0
	elif t_1 <= 1e+112:
		tmp = x / a
	elif t_1 <= 1e+253:
		tmp = x / 1.0
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z / b);
	elseif (t_1 <= -1e+41)
		tmp = Float64(x / 1.0);
	elseif (t_1 <= -2e-304)
		tmp = Float64(x / a);
	elseif (t_1 <= 4e-278)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e-112)
		tmp = Float64(x / 1.0);
	elseif (t_1 <= 1e+112)
		tmp = Float64(x / a);
	elseif (t_1 <= 1e+253)
		tmp = Float64(x / 1.0);
	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));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z / b;
	elseif (t_1 <= -1e+41)
		tmp = x / 1.0;
	elseif (t_1 <= -2e-304)
		tmp = x / a;
	elseif (t_1 <= 4e-278)
		tmp = z / b;
	elseif (t_1 <= 1e-112)
		tmp = x / 1.0;
	elseif (t_1 <= 1e+112)
		tmp = x / a;
	elseif (t_1 <= 1e+253)
		tmp = x / 1.0;
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -1e+41], N[(x / 1.0), $MachinePrecision], If[LessEqual[t$95$1, -2e-304], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, 4e-278], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e-112], N[(x / 1.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+112], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, 1e+253], N[(x / 1.0), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -1.99999999999999994e-304 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.99999999999999975e-278 or 9.9999999999999994e252 < (/.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 33.5

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

   4. Applied rewrites 33.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000001e41 or 3.99999999999999975e-278 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999995e-113 or 9.9999999999999993e111 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e252

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 19.3%

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

   if -1.00000000000000001e41 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999994e-304 or 9.9999999999999995e-113 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999993e111

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 25.9

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

   9. Applied rewrites 25.9%

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

Alternative 16: 41.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.34:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -0.34) (/ x a) (if (<= a 1.0) (/ x 1.0) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -0.34) {
		tmp = x / a;
	} else if (a <= 1.0) {
		tmp = x / 1.0;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.34d0)) then
        tmp = x / a
    else if (a <= 1.0d0) then
        tmp = x / 1.0d0
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -0.34) {
		tmp = x / a;
	} else if (a <= 1.0) {
		tmp = x / 1.0;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -0.34:
		tmp = x / a
	elif a <= 1.0:
		tmp = x / 1.0
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -0.34)
		tmp = Float64(x / a);
	elseif (a <= 1.0)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -0.34)
		tmp = x / a;
	elseif (a <= 1.0)
		tmp = x / 1.0;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -0.34], N[(x / a), $MachinePrecision], If[LessEqual[a, 1.0], N[(x / 1.0), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.340000000000000024 or 1 < a

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 25.9

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

   9. Applied rewrites 25.9%

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

   if -0.340000000000000024 < a < 1

   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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   5. 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 42.0

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

   6. Applied rewrites 42.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 19.3%

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

Alternative 17: 25.9% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / a), $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 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 42.0

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

4. Applied rewrites 42.0%

   \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
5. 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 42.0

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

6. Applied rewrites 42.0%

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

7. Taylor expanded in a around inf

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

   1. lower-/.f64 25.9

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

9. Applied rewrites 25.9%

   \[\leadsto \frac{x}{\color{blue}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025151 
(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))))