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

Percentage Accurate: 74.6% → 89.3%

Time: 5.5s

Alternatives: 15

Speedup: 0.8×

Specification

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 74.6% accurate, 1.0× speedup

Math

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

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: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, b, a - -1\right)\\ t_2 := \mathsf{fma}\left(\frac{y}{t}, \frac{z}{t\_1}, \frac{x}{t\_1}\right)\\ \mathbf{if}\;t \leq -70000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ y t) b (- a -1.0)))
        (t_2 (fma (/ y t) (/ z t_1) (/ x t_1))))
   (if (<= t -70000000000.0)
     t_2
     (if (<= t 6.5e-11) (/ (fma t x (* z y)) (fma a t (fma b y t))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), b, (a - -1.0));
	double t_2 = fma((y / t), (z / t_1), (x / t_1));
	double tmp;
	if (t <= -70000000000.0) {
		tmp = t_2;
	} else if (t <= 6.5e-11) {
		tmp = fma(t, x, (z * y)) / fma(a, t, fma(b, y, t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y / t), b, Float64(a - -1.0))
	t_2 = fma(Float64(y / t), Float64(z / t_1), Float64(x / t_1))
	tmp = 0.0
	if (t <= -70000000000.0)
		tmp = t_2;
	elseif (t <= 6.5e-11)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, fma(b, y, t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * b + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -70000000000.0], t$95$2, If[LessEqual[t, 6.5e-11], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -7e10 or 6.4999999999999995e-11 < t

   1. Initial program 74.6%

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

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 78.1%

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

   if -7e10 < t < 6.4999999999999995e-11

   1. Initial program 74.6%

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

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.1%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 69.7%

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

Alternative 2: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(\frac{y}{t}, z, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, a - -1\right)}\\ \mathbf{if}\;t \leq -80000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma (/ y t) z x) (fma (/ y t) b (- a -1.0)))))
   (if (<= t -80000000000.0)
     t_1
     (if (<= t 2.5e-35) (/ (fma t x (* z y)) (fma a t (fma b y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), z, x) / fma((y / t), b, (a - -1.0));
	double tmp;
	if (t <= -80000000000.0) {
		tmp = t_1;
	} else if (t <= 2.5e-35) {
		tmp = fma(t, x, (z * y)) / fma(a, t, fma(b, y, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / t), z, x) / fma(Float64(y / t), b, Float64(a - -1.0)))
	tmp = 0.0
	if (t <= -80000000000.0)
		tmp = t_1;
	elseif (t <= 2.5e-35)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, fma(b, y, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision] / N[(N[(y / t), $MachinePrecision] * b + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -80000000000.0], t$95$1, If[LessEqual[t, 2.5e-35], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8e10 or 2.4999999999999998e-35 < t

   1. Initial program 74.6%

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

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 74.2%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 76.3%

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

      19. lift-+.f64 N/A

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

      20. add-flip N/A

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

      21. lower--.f64 N/A

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

      22. metadata-eval 76.3%

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

   3. Applied rewrites 76.3%

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

   if -8e10 < t < 2.4999999999999998e-35

   1. Initial program 74.6%

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

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.1%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 69.7%

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

Alternative 3: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, \frac{z}{a - -1}, \frac{x}{a - -1}\right)\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, x, z \cdot y\right)}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (/ z (- a -1.0)) (/ x (- a -1.0)))))
   (if (<= t -1.9e+97)
     t_1
     (if (<= t 8.5e+51) (/ (fma t x (* z y)) (fma a t (fma b y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / t), (z / (a - -1.0)), (x / (a - -1.0)));
	double tmp;
	if (t <= -1.9e+97) {
		tmp = t_1;
	} else if (t <= 8.5e+51) {
		tmp = fma(t, x, (z * y)) / fma(a, t, fma(b, y, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y / t), Float64(z / Float64(a - -1.0)), Float64(x / Float64(a - -1.0)))
	tmp = 0.0
	if (t <= -1.9e+97)
		tmp = t_1;
	elseif (t <= 8.5e+51)
		tmp = Float64(fma(t, x, Float64(z * y)) / fma(a, t, fma(b, y, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.9e+97], t$95$1, If[LessEqual[t, 8.5e+51], N[(N[(t * x + N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9000000000000002e97 or 8.4999999999999999e51 < t

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 58.4%

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

   if -1.9000000000000002e97 < t < 8.4999999999999999e51

   1. Initial program 74.6%

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

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.1%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 69.7%

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

Alternative 4: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{y \cdot z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{t}, z \cdot y, x\right)}{1 + a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (/ (* y z) (fma a t (fma b y t)))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-316)
       (/ (fma (/ 1.0 t) (* z y) x) (+ 1.0 a))
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 5e+306) (/ (fma z (/ y t) x) (- a -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 t_2 = (y * z) / fma(a, t, fma(b, y, t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-316) {
		tmp = fma((1.0 / t), (z * y), x) / (1.0 + a);
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+306) {
		tmp = fma(z, (y / t), x) / (a - -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)))
	t_2 = Float64(Float64(y * z) / fma(a, t, fma(b, y, t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-316)
		tmp = Float64(fma(Float64(1.0 / t), Float64(z * y), x) / Float64(1.0 + a));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+306)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a - -1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * z), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-316], N[(N[(N[(1.0 / t), $MachinePrecision] * N[(z * y), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+306], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -5.0000000171117013e-316 < (/.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 74.6%

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

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.1%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 69.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 42.1%

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

   6. Applied rewrites 42.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.0000000171117013e-316

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999999e306

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 57.8%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 57.8%

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

   6. Applied rewrites 57.8%

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

   if 4.9999999999999999e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 5: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{y \cdot z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ (* y z) (fma a t (fma b y t)))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-316)
       (/ t_1 (+ 1.0 a))
       (if (<= t_2 0.0)
         t_3
         (if (<= t_2 5e+306) (/ (fma z (/ y t) x) (- a -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);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = (y * z) / fma(a, t, fma(b, y, t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-316) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 5e+306) {
		tmp = fma(z, (y / t), x) / (a - -1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(Float64(y * z) / fma(a, t, fma(b, y, t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-316)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 5e+306)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a - -1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * z), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-316], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 5e+306], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or -5.0000000171117013e-316 < (/.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 74.6%

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

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.1%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 69.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 42.1%

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

   6. Applied rewrites 42.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.0000000171117013e-316

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999999e306

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 57.8%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 57.8%

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

   6. Applied rewrites 57.8%

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

   if 4.9999999999999999e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 6: 72.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \mathsf{fma}\left(\frac{y}{t}, \frac{z}{a - -1}, \frac{x}{a - -1}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-322}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(b, y, t\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (fma (/ y t) (/ z (- a -1.0)) (/ x (- a -1.0)))))
   (if (<= t_1 -5e-316)
     t_2
     (if (<= t_1 5e-322)
       (/ (* y z) (fma a t (fma b y t)))
       (if (<= t_1 5e+306) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = fma((y / t), (z / (a - -1.0)), (x / (a - -1.0)));
	double tmp;
	if (t_1 <= -5e-316) {
		tmp = t_2;
	} else if (t_1 <= 5e-322) {
		tmp = (y * z) / fma(a, t, fma(b, y, t));
	} else if (t_1 <= 5e+306) {
		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 = fma(Float64(y / t), Float64(z / Float64(a - -1.0)), Float64(x / Float64(a - -1.0)))
	tmp = 0.0
	if (t_1 <= -5e-316)
		tmp = t_2;
	elseif (t_1 <= 5e-322)
		tmp = Float64(Float64(y * z) / fma(a, t, fma(b, y, t)));
	elseif (t_1 <= 5e+306)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / t), $MachinePrecision] * N[(z / N[(a - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-316], t$95$2, If[LessEqual[t$95$1, 5e-322], N[(N[(y * z), $MachinePrecision] / N[(a * t + N[(b * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], 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))) < -5.0000000171117013e-316 or 4.9900630229965901e-322 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999999e306

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 58.4%

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

   if -5.0000000171117013e-316 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9900630229965901e-322

   1. Initial program 74.6%

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

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

      4. add-to-fraction N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. distribute-rgt-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 64.1%

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

      19. lower-unsound-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 69.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 42.1%

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

   6. Applied rewrites 42.1%

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

   if 4.9999999999999999e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 7: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a - -1}\\ \mathbf{elif}\;t \leq 1.58 \cdot 10^{-18}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.2e-43)
   (/ (fma (/ z t) y x) (- a -1.0))
   (if (<= t 1.58e-18)
     (/ (+ z (/ (* t x) y)) b)
     (/ (fma z (/ y t) x) (- a -1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.2e-43) {
		tmp = fma((z / t), y, x) / (a - -1.0);
	} else if (t <= 1.58e-18) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = fma(z, (y / t), x) / (a - -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.2e-43)
		tmp = Float64(fma(Float64(z / t), y, x) / Float64(a - -1.0));
	elseif (t <= 1.58e-18)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	else
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a - -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.2e-43], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.58e-18], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2000000000000001e-43

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   6. Applied rewrites 56.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

   8. Applied rewrites 56.2%

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

   if -1.2000000000000001e-43 < t < 1.5800000000000001e-18

   1. Initial program 74.6%

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

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 78.1%

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

   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 40.8%

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

   6. Applied rewrites 40.8%

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

   if 1.5800000000000001e-18 < t

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 57.8%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 57.8%

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

   6. Applied rewrites 57.8%

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

Alternative 8: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a - -1}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.58 \cdot 10^{-18}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma z (/ y t) x) (- a -1.0))))
   (if (<= t -1.2e-43)
     t_1
     (if (<= t 1.58e-18) (/ (+ 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, (y / t), x) / (a - -1.0);
	double tmp;
	if (t <= -1.2e-43) {
		tmp = t_1;
	} else if (t <= 1.58e-18) {
		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(z, Float64(y / t), x) / Float64(a - -1.0))
	tmp = 0.0
	if (t <= -1.2e-43)
		tmp = t_1;
	elseif (t <= 1.58e-18)
		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[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2e-43], t$95$1, If[LessEqual[t, 1.58e-18], 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 < -1.2000000000000001e-43 or 1.5800000000000001e-18 < t

   1. Initial program 74.6%

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

   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 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 57.8%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lift--.f64 57.8%

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

   6. Applied rewrites 57.8%

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

   if -1.2000000000000001e-43 < t < 1.5800000000000001e-18

   1. Initial program 74.6%

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

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 78.1%

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

   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 40.8%

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

   6. Applied rewrites 40.8%

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

Alternative 9: 68.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{y}{t}, b, a - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]

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 5e+306) (/ x (fma (/ y t) b (- a -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 <= 5e+306) {
		tmp = x / fma((y / t), b, (a - -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 <= 5e+306)
		tmp = Float64(x / fma(Float64(y / t), b, Float64(a - -1.0)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], N[(x / N[(N[(y / t), $MachinePrecision] * b + N[(a - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 4.9999999999999999e306 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999999e306

   1. Initial program 74.6%

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

   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 51.9%

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

   4. Applied rewrites 51.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. add-flip N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. sub-flip-reverse N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. sub-negate N/A

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

      17. sub-negate-rev N/A

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

      18. add-flip N/A

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

      19. +-commutative N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. add-flip N/A

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

      23. metadata-eval N/A

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

      24. lift--.f64 53.3%

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

   6. Applied rewrites 53.3%

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

Alternative 10: 60.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\left(1 - \frac{-1}{a}\right) \cdot a}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{z + \frac{t \cdot x}{y}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.2e-43)
   (/ x (* (- 1.0 (/ -1.0 a)) a))
   (if (<= t 1.5e+52) (/ (+ z (/ (* t x) y)) b) (/ x (- a -1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.2e-43) {
		tmp = x / ((1.0 - (-1.0 / a)) * a);
	} else if (t <= 1.5e+52) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = x / (a - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-1.2d-43)) then
        tmp = x / ((1.0d0 - ((-1.0d0) / a)) * a)
    else if (t <= 1.5d+52) then
        tmp = (z + ((t * x) / y)) / b
    else
        tmp = x / (a - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.2e-43) {
		tmp = x / ((1.0 - (-1.0 / a)) * a);
	} else if (t <= 1.5e+52) {
		tmp = (z + ((t * x) / y)) / b;
	} else {
		tmp = x / (a - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.2e-43:
		tmp = x / ((1.0 - (-1.0 / a)) * a)
	elif t <= 1.5e+52:
		tmp = (z + ((t * x) / y)) / b
	else:
		tmp = x / (a - -1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.2e-43)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-1.0 / a)) * a));
	elseif (t <= 1.5e+52)
		tmp = Float64(Float64(z + Float64(Float64(t * x) / y)) / b);
	else
		tmp = Float64(x / Float64(a - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.2e-43)
		tmp = x / ((1.0 - (-1.0 / a)) * a);
	elseif (t <= 1.5e+52)
		tmp = (z + ((t * x) / y)) / b;
	else
		tmp = x / (a - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.2e-43], N[(x / N[(N[(1.0 - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e+52], N[(N[(z + N[(N[(t * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2000000000000001e-43

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   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. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

         \[\leadsto \frac{x}{\left(1 - \frac{-1}{a}\right) \cdot a} \]

      8. lower-unsound-/.f64 41.8%

         \[\leadsto \frac{x}{\left(1 - \frac{-1}{a}\right) \cdot a} \]

   6. Applied rewrites 41.8%

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

   if -1.2000000000000001e-43 < t < 1.5e52

   1. Initial program 74.6%

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

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 78.1%

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

   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 40.8%

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

   6. Applied rewrites 40.8%

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

   if 1.5e52 < t

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   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 41.9%

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

   6. Applied rewrites 41.9%

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

Alternative 11: 55.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\left(1 - \frac{-1}{a}\right) \cdot a}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a - -1}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.2e-43)
   (/ x (* (- 1.0 (/ -1.0 a)) a))
   (if (<= t 2.6e+52) (/ z b) (/ x (- a -1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.2e-43) {
		tmp = x / ((1.0 - (-1.0 / a)) * a);
	} else if (t <= 2.6e+52) {
		tmp = z / b;
	} else {
		tmp = x / (a - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-1.2d-43)) then
        tmp = x / ((1.0d0 - ((-1.0d0) / a)) * a)
    else if (t <= 2.6d+52) then
        tmp = z / b
    else
        tmp = x / (a - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.2e-43) {
		tmp = x / ((1.0 - (-1.0 / a)) * a);
	} else if (t <= 2.6e+52) {
		tmp = z / b;
	} else {
		tmp = x / (a - -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -1.2e-43:
		tmp = x / ((1.0 - (-1.0 / a)) * a)
	elif t <= 2.6e+52:
		tmp = z / b
	else:
		tmp = x / (a - -1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.2e-43)
		tmp = Float64(x / Float64(Float64(1.0 - Float64(-1.0 / a)) * a));
	elseif (t <= 2.6e+52)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / Float64(a - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -1.2e-43)
		tmp = x / ((1.0 - (-1.0 / a)) * a);
	elseif (t <= 2.6e+52)
		tmp = z / b;
	else
		tmp = x / (a - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.2e-43], N[(x / N[(N[(1.0 - N[(-1.0 / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e+52], N[(z / b), $MachinePrecision], N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2000000000000001e-43

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   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. sub-to-mult N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound--.f64 N/A

         \[\leadsto \frac{x}{\left(1 - \frac{-1}{a}\right) \cdot a} \]

      8. lower-unsound-/.f64 41.8%

         \[\leadsto \frac{x}{\left(1 - \frac{-1}{a}\right) \cdot a} \]

   6. Applied rewrites 41.8%

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

   if -1.2000000000000001e-43 < t < 2.6e52

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

   if 2.6e52 < t

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   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 41.9%

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

   6. Applied rewrites 41.9%

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

Alternative 12: 55.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{a - -1}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- a -1.0))))
   (if (<= t -1.2e-43) t_1 (if (<= t 2.6e+52) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a - -1.0);
	double tmp;
	if (t <= -1.2e-43) {
		tmp = t_1;
	} else if (t <= 2.6e+52) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (a - (-1.0d0))
    if (t <= (-1.2d-43)) then
        tmp = t_1
    else if (t <= 2.6d+52) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a - -1.0);
	double tmp;
	if (t <= -1.2e-43) {
		tmp = t_1;
	} else if (t <= 2.6e+52) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a - -1.0)
	tmp = 0
	if t <= -1.2e-43:
		tmp = t_1
	elif t <= 2.6e+52:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a - -1.0))
	tmp = 0.0
	if (t <= -1.2e-43)
		tmp = t_1;
	elseif (t <= 2.6e+52)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (a - -1.0);
	tmp = 0.0;
	if (t <= -1.2e-43)
		tmp = t_1;
	elseif (t <= 2.6e+52)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2e-43], t$95$1, If[LessEqual[t, 2.6e+52], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2000000000000001e-43 or 2.6e52 < t

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]
   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 41.9%

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

   6. Applied rewrites 41.9%

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

   if -1.2000000000000001e-43 < t < 2.6e52

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 13: 43.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.05e+27) (/ x a) (if (<= a 4.4e+15) (/ z b) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.05e+27) {
		tmp = x / a;
	} else if (a <= 4.4e+15) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.05d+27)) then
        tmp = x / a
    else if (a <= 4.4d+15) then
        tmp = z / b
    else
        tmp = x / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.05e+27) {
		tmp = x / a;
	} else if (a <= 4.4e+15) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.05e+27:
		tmp = x / a
	elif a <= 4.4e+15:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.05e+27)
		tmp = Float64(x / a);
	elseif (a <= 4.4e+15)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -2.05e+27)
		tmp = x / a;
	elseif (a <= 4.4e+15)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.05e+27], N[(x / a), $MachinePrecision], If[LessEqual[a, 4.4e+15], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.0500000000000001e27 or 4.4e15 < a

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 25.6%

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

   7. Applied rewrites 25.6%

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

   if -2.0500000000000001e27 < a < 4.4e15

   1. Initial program 74.6%

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

   2. Taylor expanded in y around inf

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

      1. lower-/.f64 34.4%

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

   4. Applied rewrites 34.4%

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

Alternative 14: 40.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 460000000000:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -2.05e+27) (/ x a) (if (<= a 460000000000.0) (/ x 1.0) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -2.05e+27) {
		tmp = x / a;
	} else if (a <= 460000000000.0) {
		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 <= (-2.05d+27)) then
        tmp = x / a
    else if (a <= 460000000000.0d0) 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 <= -2.05e+27) {
		tmp = x / a;
	} else if (a <= 460000000000.0) {
		tmp = x / 1.0;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -2.05e+27:
		tmp = x / a
	elif a <= 460000000000.0:
		tmp = x / 1.0
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -2.05e+27)
		tmp = Float64(x / a);
	elseif (a <= 460000000000.0)
		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 <= -2.05e+27)
		tmp = x / a;
	elseif (a <= 460000000000.0)
		tmp = x / 1.0;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -2.05e+27], N[(x / a), $MachinePrecision], If[LessEqual[a, 460000000000.0], N[(x / 1.0), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.0500000000000001e27 or 4.6e11 < a

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 25.6%

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

   7. Applied rewrites 25.6%

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

   if -2.0500000000000001e27 < a < 4.6e11

   1. Initial program 74.6%

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

   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 41.9%

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 19.4%

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

Alternative 15: 25.6% accurate, 5.8× speedup

Math

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

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 74.6%

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

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 41.9%

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

4. Applied rewrites 41.9%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 25.6%

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

7. Applied rewrites 25.6%

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

Reproduce

herbie shell --seed 2025188 
(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))))