Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Percentage Accurate: 91.3% → 95.7%

Time: 3.1s

Alternatives: 10

Speedup: 0.7×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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)
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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

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)
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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.5 \cdot x, \frac{y}{a \cdot z}, \frac{t}{a} \cdot -4.5\right) \cdot z\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+262}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (fma (* 0.5 x) (/ y (* a z)) (* (/ t a) -4.5)) z))
        (t_2 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 1e+262) (/ (fma (* -9.0 z) t (* y x)) (+ a a)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((0.5 * x), (y / (a * z)), ((t / a) * -4.5)) * z;
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+262) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(fma(Float64(0.5 * x), Float64(y / Float64(a * z)), Float64(Float64(t / a) * -4.5)) * z)
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+262)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(0.5 * x), $MachinePrecision] * N[(y / N[(a * z), $MachinePrecision]), $MachinePrecision] + N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+262], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -inf.0 or 1e262 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 70.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      4. associate-/l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(x \cdot \frac{y}{a \cdot z}\right) + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot x\right) \cdot \frac{y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot z}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot z}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot z}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot z}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot z}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot z}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      12. lower-/.f64 87.0

          \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \frac{y}{a \cdot z}, \frac{t}{a} \cdot -4.5\right) \cdot z \]

   4. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{y}{a \cdot z}, \frac{t}{a} \cdot -4.5\right) \cdot z} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1e262

   1. Initial program 98.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 480000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= a 480000000000.0)
   (/ (fma (* -9.0 z) t (* y x)) (+ a a))
   (- (* y (/ x (+ a a))) (* (* 9.0 z) (/ t (+ a a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 480000000000.0) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a + a);
	} else {
		tmp = (y * (x / (a + a))) - ((9.0 * z) * (t / (a + a)));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 480000000000.0)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a + a));
	else
		tmp = Float64(Float64(y * Float64(x / Float64(a + a))) - Float64(Float64(9.0 * z) * Float64(t / Float64(a + a))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[a, 480000000000.0], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(9.0 * z), $MachinePrecision] * N[(t / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.8e11

   1. Initial program 93.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 93.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 93.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   if 4.8e11 < a

   1. Initial program 84.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      10. lower-/.f64 N/A

          \[\leadsto y \cdot \color{blue}{\frac{x}{a \cdot 2}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      11. *-commutative N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{2 \cdot a}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      12. count-2-rev N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{a + a}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      13. lower-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\color{blue}{a + a}} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      15. lift-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \frac{\color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      16. *-commutative N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \frac{\color{blue}{\left(9 \cdot z\right)} \cdot t}{a \cdot 2} \]

      17. associate-/l* N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \color{blue}{\left(9 \cdot z\right) \cdot \frac{t}{a \cdot 2}} \]

      18. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \color{blue}{\left(9 \cdot z\right) \cdot \frac{t}{a \cdot 2}} \]

      19. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \color{blue}{\left(9 \cdot z\right)} \cdot \frac{t}{a \cdot 2} \]

      20. lower-/.f64 N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \color{blue}{\frac{t}{a \cdot 2}} \]

      21. *-commutative N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{\color{blue}{2 \cdot a}} \]

      22. count-2-rev N/A

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{\color{blue}{a + a}} \]

      23. lower-+.f64 92.0

          \[\leadsto y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{\color{blue}{a + a}} \]

   3. Applied rewrites 92.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{a + a} - \left(9 \cdot z\right) \cdot \frac{t}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.5 \cdot x, \frac{y}{a \cdot t}, \frac{z}{a} \cdot -4.5\right) \cdot t\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+213}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (fma (* 0.5 x) (/ y (* a t)) (* (/ z a) -4.5)) t))
        (t_2 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 2e+213) (/ (fma (* -9.0 z) t (* y x)) (+ a a)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((0.5 * x), (y / (a * t)), ((z / a) * -4.5)) * t;
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+213) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(fma(Float64(0.5 * x), Float64(y / Float64(a * t)), Float64(Float64(z / a) * -4.5)) * t)
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+213)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(0.5 * x), $MachinePrecision] * N[(y / N[(a * t), $MachinePrecision]), $MachinePrecision] + N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+213], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -inf.0 or 1.99999999999999997e213 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 74.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right) \cdot \color{blue}{t} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t} + \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      4. associate-/l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(x \cdot \frac{y}{a \cdot t}\right) + \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{2} \cdot x\right) \cdot \frac{y}{a \cdot t} + \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot t}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot t}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot t}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot t}, \frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot t}, \frac{z}{a} \cdot \frac{-9}{2}\right) \cdot t \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, \frac{y}{a \cdot t}, \frac{z}{a} \cdot \frac{-9}{2}\right) \cdot t \]

      12. lower-/.f64 84.8

          \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \frac{y}{a \cdot t}, \frac{z}{a} \cdot -4.5\right) \cdot t \]

   4. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{y}{a \cdot t}, \frac{z}{a} \cdot -4.5\right) \cdot t} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.99999999999999997e213

   1. Initial program 98.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot y}{z}, 0.5, -4.5 \cdot t\right)}{a} \cdot z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma (/ (* x y) z) 0.5 (* -4.5 t)) a) z)
     (if (<= t_1 4e+244)
       (/ (fma (* -9.0 z) t (* y x)) (+ a a))
       (* (* z (/ t a)) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(((x * y) / z), 0.5, (-4.5 * t)) / a) * z;
	} else if (t_1 <= 4e+244) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(x * y) / z), 0.5, Float64(-4.5 * t)) / a) * z);
	elseif (t_1 <= 4e+244)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a + a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] * 0.5 + N[(-4.5 * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 4e+244], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 64.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 66.9

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot y}{a \cdot z} \cdot \frac{1}{2} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      12. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z \]

   7. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-9}{2} \cdot t + \frac{1}{2} \cdot \frac{x \cdot y}{z}}{a} \cdot z \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot t + \frac{1}{2} \cdot \frac{x \cdot y}{z}}{a} \cdot z \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{-9}{2} \cdot t}{a} \cdot z \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{x \cdot y}{z} \cdot \frac{1}{2} + \frac{-9}{2} \cdot t}{a} \cdot z \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot y}{z}, \frac{1}{2}, \frac{-9}{2} \cdot t\right)}{a} \cdot z \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot y}{z}, \frac{1}{2}, \frac{-9}{2} \cdot t\right)}{a} \cdot z \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot y}{z}, \frac{1}{2}, \frac{-9}{2} \cdot t\right)}{a} \cdot z \]

      7. lower-*.f64 93.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot y}{z}, 0.5, -4.5 \cdot t\right)}{a} \cdot z \]

   10. Applied rewrites 93.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{x \cdot y}{z}, 0.5, -4.5 \cdot t\right)}{a} \cdot z \]

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.0000000000000003e244

   1. Initial program 95.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   if 4.0000000000000003e244 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 73.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 73.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      9. lift-*.f64 73.3

         \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{a} \]

   6. Applied rewrites 73.3%

      \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{\color{blue}{a}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      8. lower-/.f64 73.3

         \[\leadsto \left(z \cdot t\right) \cdot \frac{-4.5}{\color{blue}{a}} \]

   8. Applied rewrites 73.3%

      \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{-4.5}{a}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\frac{-9}{2}}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      6. lift-*.f64 73.3

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{-4.5} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      9. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      11. lift-/.f64 92.5

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]

   10. Applied rewrites 92.5%

       \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right) \cdot -4.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (* (/ t a) -4.5) z)
     (if (<= t_1 4e+244)
       (/ (fma (* -9.0 z) t (* y x)) (+ a a))
       (* (* z (/ t a)) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t / a) * -4.5) * z;
	} else if (t_1 <= 4e+244) {
		tmp = fma((-9.0 * z), t, (y * x)) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	elseif (t_1 <= 4e+244)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(a + a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 4e+244], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 64.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 66.9

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot y}{a \cdot z} \cdot \frac{1}{2} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      12. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z \]

   7. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      3. lift-*.f64 94.1

         \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   10. Applied rewrites 94.1%

       \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.0000000000000003e244

   1. Initial program 95.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   if 4.0000000000000003e244 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 73.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 73.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      9. lift-*.f64 73.3

         \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{a} \]

   6. Applied rewrites 73.3%

      \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{\color{blue}{a}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      8. lower-/.f64 73.3

         \[\leadsto \left(z \cdot t\right) \cdot \frac{-4.5}{\color{blue}{a}} \]

   8. Applied rewrites 73.3%

      \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{-4.5}{a}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\frac{-9}{2}}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      6. lift-*.f64 73.3

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{-4.5} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      9. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      11. lift-/.f64 92.5

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]

   10. Applied rewrites 92.5%

       \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right) \cdot -4.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+244}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot -9\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* (* (/ t a) -4.5) z)
     (if (<= t_1 4e+244)
       (/ (fma y x (* (* t z) -9.0)) (+ a a))
       (* (* z (/ t a)) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((t / a) * -4.5) * z;
	} else if (t_1 <= 4e+244) {
		tmp = fma(y, x, ((t * z) * -9.0)) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	elseif (t_1 <= 4e+244)
		tmp = Float64(fma(y, x, Float64(Float64(t * z) * -9.0)) / Float64(a + a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 4e+244], N[(N[(y * x + N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 64.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 66.9

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot y}{a \cdot z} \cdot \frac{1}{2} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      12. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z \]

   7. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      3. lift-*.f64 94.1

         \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   10. Applied rewrites 94.1%

       \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.0000000000000003e244

   1. Initial program 95.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, y \cdot x\right)}{a + a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a + a} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-9 \cdot z\right) \cdot t + y \cdot x}}{a + a} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{-9 \cdot \left(z \cdot t\right)} + y \cdot x}{a + a} \]

      5. *-commutative N/A

         \[\leadsto \frac{-9 \cdot \color{blue}{\left(t \cdot z\right)} + y \cdot x}{a + a} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x + -9 \cdot \left(t \cdot z\right)}}{a + a} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, -9 \cdot \left(t \cdot z\right)\right)}}{a + a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}{a + a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(t \cdot z\right) \cdot -9}\right)}{a + a} \]

      10. lift-*.f64 95.1

          \[\leadsto \frac{\mathsf{fma}\left(y, x, \color{blue}{\left(t \cdot z\right)} \cdot -9\right)}{a + a} \]

   5. Applied rewrites 95.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot -9\right)}}{a + a} \]

   if 4.0000000000000003e244 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 73.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 73.3

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      9. lift-*.f64 73.3

         \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{a} \]

   6. Applied rewrites 73.3%

      \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{\color{blue}{a}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      8. lower-/.f64 73.3

         \[\leadsto \left(z \cdot t\right) \cdot \frac{-4.5}{\color{blue}{a}} \]

   8. Applied rewrites 73.3%

      \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{-4.5}{a}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\frac{-9}{2}}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      6. lift-*.f64 73.3

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{-4.5} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      9. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      11. lift-/.f64 92.5

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]

   10. Applied rewrites 92.5%

       \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right) \cdot -4.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\left(t \cdot \frac{z}{a}\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{t}{a}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -6e+50)
     (* (* t (/ z a)) -4.5)
     (if (<= t_1 5e+41) (/ (* x y) (+ a a)) (* (* z (/ t a)) -4.5)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -6e+50) {
		tmp = (t * (z / a)) * -4.5;
	} else if (t_1 <= 5e+41) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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)
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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-6d+50)) then
        tmp = (t * (z / a)) * (-4.5d0)
    else if (t_1 <= 5d+41) then
        tmp = (x * y) / (a + a)
    else
        tmp = (z * (t / a)) * (-4.5d0)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -6e+50) {
		tmp = (t * (z / a)) * -4.5;
	} else if (t_1 <= 5e+41) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = (z * (t / a)) * -4.5;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -6e+50:
		tmp = (t * (z / a)) * -4.5
	elif t_1 <= 5e+41:
		tmp = (x * y) / (a + a)
	else:
		tmp = (z * (t / a)) * -4.5
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -6e+50)
		tmp = Float64(Float64(t * Float64(z / a)) * -4.5);
	elseif (t_1 <= 5e+41)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(Float64(z * Float64(t / a)) * -4.5);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -6e+50)
		tmp = (t * (z / a)) * -4.5;
	elseif (t_1 <= 5e+41)
		tmp = (x * y) / (a + a);
	else
		tmp = (z * (t / a)) * -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -6e+50], N[(N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, 5e+41], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.9999999999999996e50

   1. Initial program 86.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 72.8

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot \frac{-9}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot \frac{-9}{2} \]

      5. lower-/.f64 77.6

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot -4.5 \]

   6. Applied rewrites 77.6%

      \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot -4.5 \]

   if -5.9999999999999996e50 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000022e41

   1. Initial program 94.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 94.8

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 94.8

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 71.3

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

   6. Applied rewrites 71.3%

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

   if 5.00000000000000022e41 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 72.1

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-9}{2} \cdot \left(t \cdot z\right)}{\color{blue}{a}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      9. lift-*.f64 72.1

         \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{a} \]

   6. Applied rewrites 72.1%

      \[\leadsto \frac{\left(t \cdot z\right) \cdot -4.5}{\color{blue}{a}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(t \cdot z\right) \cdot \frac{-9}{2}}{a} \]

      4. associate-/l* N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\color{blue}{\frac{-9}{2}}}{a} \]

      8. lower-/.f64 72.1

         \[\leadsto \left(z \cdot t\right) \cdot \frac{-4.5}{\color{blue}{a}} \]

   8. Applied rewrites 72.1%

      \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{-4.5}{a}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \color{blue}{\frac{\frac{-9}{2}}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(z \cdot t\right) \cdot \frac{\frac{-9}{2}}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{-9}{2}}{\color{blue}{a}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      6. lift-*.f64 72.1

         \[\leadsto \frac{z \cdot t}{a} \cdot \color{blue}{-4.5} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot t}{a} \cdot \frac{-9}{2} \]

      9. associate-/l* N/A

         \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot \frac{-9}{2} \]

      11. lift-/.f64 76.8

          \[\leadsto \left(z \cdot \frac{t}{a}\right) \cdot -4.5 \]

   10. Applied rewrites 76.8%

       \[\leadsto \color{blue}{\left(z \cdot \frac{t}{a}\right) \cdot -4.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\left(t \cdot \frac{z}{a}\right) \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -6e+50)
     (* (* t (/ z a)) -4.5)
     (if (<= t_1 5e+41) (/ (* x y) (+ a a)) (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -6e+50) {
		tmp = (t * (z / a)) * -4.5;
	} else if (t_1 <= 5e+41) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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)
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) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-6d+50)) then
        tmp = (t * (z / a)) * (-4.5d0)
    else if (t_1 <= 5d+41) then
        tmp = (x * y) / (a + a)
    else
        tmp = ((t / a) * (-4.5d0)) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -6e+50) {
		tmp = (t * (z / a)) * -4.5;
	} else if (t_1 <= 5e+41) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -6e+50:
		tmp = (t * (z / a)) * -4.5
	elif t_1 <= 5e+41:
		tmp = (x * y) / (a + a)
	else:
		tmp = ((t / a) * -4.5) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -6e+50)
		tmp = Float64(Float64(t * Float64(z / a)) * -4.5);
	elseif (t_1 <= 5e+41)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -6e+50)
		tmp = (t * (z / a)) * -4.5;
	elseif (t_1 <= 5e+41)
		tmp = (x * y) / (a + a);
	else
		tmp = ((t / a) * -4.5) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -6e+50], N[(N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[t$95$1, 5e+41], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.9999999999999996e50

   1. Initial program 86.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 72.8

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      3. associate-/l* N/A

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot \frac{-9}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot \frac{-9}{2} \]

      5. lower-/.f64 77.6

         \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot -4.5 \]

   6. Applied rewrites 77.6%

      \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot -4.5 \]

   if -5.9999999999999996e50 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000022e41

   1. Initial program 94.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 94.8

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 94.8

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 71.3

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

   6. Applied rewrites 71.3%

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

   if 5.00000000000000022e41 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 72.1

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot y}{a \cdot z} \cdot \frac{1}{2} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      12. lower-/.f64 87.3

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z \]

   7. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      3. lift-*.f64 76.7

         \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   10. Applied rewrites 76.7%

       \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ t_2 := \left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)) (t_2 (* (* (/ t a) -4.5) z)))
   (if (<= t_1 -1e-61) t_2 (if (<= t_1 5e+41) (/ (* x y) (+ a a)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = ((t / a) * -4.5) * z;
	double tmp;
	if (t_1 <= -1e-61) {
		tmp = t_2;
	} else if (t_1 <= 5e+41) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    t_2 = ((t / a) * (-4.5d0)) * z
    if (t_1 <= (-1d-61)) then
        tmp = t_2
    else if (t_1 <= 5d+41) then
        tmp = (x * y) / (a + a)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = ((t / a) * -4.5) * z;
	double tmp;
	if (t_1 <= -1e-61) {
		tmp = t_2;
	} else if (t_1 <= 5e+41) {
		tmp = (x * y) / (a + a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	t_2 = ((t / a) * -4.5) * z
	tmp = 0
	if t_1 <= -1e-61:
		tmp = t_2
	elif t_1 <= 5e+41:
		tmp = (x * y) / (a + a)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	t_2 = Float64(Float64(Float64(t / a) * -4.5) * z)
	tmp = 0.0
	if (t_1 <= -1e-61)
		tmp = t_2;
	elseif (t_1 <= 5e+41)
		tmp = Float64(Float64(x * y) / Float64(a + a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	t_2 = ((t / a) * -4.5) * z;
	tmp = 0.0;
	if (t_1 <= -1e-61)
		tmp = t_2;
	elseif (t_1 <= 5e+41)
		tmp = (x * y) / (a + a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-61], t$95$2, If[LessEqual[t$95$1, 5e+41], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1e-61 or 5.00000000000000022e41 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 88.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \color{blue}{\frac{-9}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{t \cdot z}{a} \cdot \frac{-9}{2} \]

      4. lower-*.f64 68.4

         \[\leadsto \frac{t \cdot z}{a} \cdot -4.5 \]

   4. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z}\right) \cdot \color{blue}{z} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \frac{x \cdot y}{a \cdot z} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot y}{a \cdot z} \cdot \frac{1}{2} + \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, \frac{1}{2}, \frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      12. lower-/.f64 85.2

          \[\leadsto \mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z \]

   7. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{y}{a \cdot z}, 0.5, \frac{t}{a} \cdot -4.5\right) \cdot z} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{-9}{2} \cdot \frac{t}{a}\right) \cdot z \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{t}{a} \cdot \frac{-9}{2}\right) \cdot z \]

      3. lift-*.f64 71.3

         \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   10. Applied rewrites 71.3%

       \[\leadsto \left(\frac{t}{a} \cdot -4.5\right) \cdot z \]

   if -1e-61 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000022e41

   1. Initial program 94.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      13. lower-*.f64 94.7

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

      16. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

      17. lower-+.f64 94.7

          \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   3. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

   4. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 75.2

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

   6. Applied rewrites 75.2%

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

Alternative 10: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{x \cdot y}{a + a} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (/ (* x y) (+ a a)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (x * y) / (a + a);
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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)
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
    code = (x * y) / (a + a)
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return (x * y) / (a + a);
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (x * y) / (a + a)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(x * y) / Float64(a + a))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (x * y) / (a + a);
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t}}{a \cdot 2} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t + x \cdot y}}{a \cdot 2} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{9 \cdot z}\right)\right) \cdot t + x \cdot y}{a \cdot 2} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(9\right)\right) \cdot z\right)} \cdot t + x \cdot y}{a \cdot 2} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\left(\color{blue}{-9} \cdot z\right) \cdot t + x \cdot y}{a \cdot 2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-9 \cdot z, t, x \cdot y\right)}}{a \cdot 2} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{-9 \cdot z}, t, x \cdot y\right)}{a \cdot 2} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   13. lower-*.f64 91.5

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, \color{blue}{y \cdot x}\right)}{a \cdot 2} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a \cdot 2}} \]

   15. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{2 \cdot a}} \]

   16. count-2-rev N/A

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

   17. lower-+.f64 91.5

       \[\leadsto \frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{\color{blue}{a + a}} \]

3. Applied rewrites 91.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{a + a}} \]

4. Taylor expanded in x around inf

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. lower-*.f64 50.7

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

6. Applied rewrites 50.7%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{a + a} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64
  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))