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

Percentage Accurate: 91.3% → 93.9%

Time: 4.4s

Alternatives: 11

Speedup: 0.7×

• 28.8% 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: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a\_m \cdot 2} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot \frac{y}{t}, 0.5, -4.5 \cdot z\right)}{a\_m} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot -9\right)}{a\_m + a\_m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (- (* x y) (* (* z 9.0) t)) (* a_m 2.0)) (- INFINITY))
    (* (/ (fma (* x (/ y t)) 0.5 (* -4.5 z)) a_m) t)
    (/ (fma y x (* (* z t) -9.0)) (+ a_m a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if ((((x * y) - ((z * 9.0) * t)) / (a_m * 2.0)) <= -((double) INFINITY)) {
		tmp = (fma((x * (y / t)), 0.5, (-4.5 * z)) / a_m) * t;
	} else {
		tmp = fma(y, x, ((z * t) * -9.0)) / (a_m + a_m);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a_m * 2.0)) <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(x * Float64(y / t)), 0.5, Float64(-4.5 * z)) / a_m) * t);
	else
		tmp = Float64(fma(y, x, Float64(Float64(z * t) * -9.0)) / Float64(a_m + a_m));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-4.5 * z), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x + N[(N[(z * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 89.0

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

   8. Applied rewrites 89.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y \cdot x}{t}, 0.5, -4.5 \cdot z\right)}{a} \cdot t \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.2

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

   10. Applied rewrites 92.2%

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

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

   1. Initial program 93.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.1

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

   6. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 94.3

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

   8. Applied rewrites 94.3%

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

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+294}:\\ \;\;\;\;\frac{-9 \cdot \left(t \cdot z\right)}{a\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{z}{a\_m}\right) \cdot -4.5\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_1 -2e+19)
      (/ (* (* -4.5 t) z) a_m)
      (if (<= t_1 5e-88)
        (/ (* x y) (+ a_m a_m))
        (if (<= t_1 1e+294)
          (/ (* -9.0 (* t z)) (* a_m 2.0))
          (* (* t (/ z a_m)) -4.5)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 1e+294) {
		tmp = (-9.0 * (t * z)) / (a_m * 2.0);
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-2d+19)) then
        tmp = (((-4.5d0) * t) * z) / a_m
    else if (t_1 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else if (t_1 <= 1d+294) then
        tmp = ((-9.0d0) * (t * z)) / (a_m * 2.0d0)
    else
        tmp = (t * (z / a_m)) * (-4.5d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 1e+294) {
		tmp = (-9.0 * (t * z)) / (a_m * 2.0);
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -2e+19:
		tmp = ((-4.5 * t) * z) / a_m
	elif t_1 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	elif t_1 <= 1e+294:
		tmp = (-9.0 * (t * z)) / (a_m * 2.0)
	else:
		tmp = (t * (z / a_m)) * -4.5
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a_m);
	elseif (t_1 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (t_1 <= 1e+294)
		tmp = Float64(Float64(-9.0 * Float64(t * z)) / Float64(a_m * 2.0));
	else
		tmp = Float64(Float64(t * Float64(z / a_m)) * -4.5);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -2e+19)
		tmp = ((-4.5 * t) * z) / a_m;
	elseif (t_1 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	elseif (t_1 <= 1e+294)
		tmp = (-9.0 * (t * z)) / (a_m * 2.0);
	else
		tmp = (t * (z / a_m)) * -4.5;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -2e+19], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+294], N[(N[(-9.0 * N[(t * z), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2e19

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 72.3

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

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 72.3

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

   7. Applied rewrites 72.3%

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

   if -2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.00000000000000007e294

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 1.00000000000000007e294 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 70.5

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

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. lift-/.f64 95.4

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

   7. Applied rewrites 95.4%

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

Alternative 3: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;t\_1 \leq 10^{+294}:\\ \;\;\;\;\frac{\left(z \cdot t\right) \cdot -4.5}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{z}{a\_m}\right) \cdot -4.5\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_1 -2e+19)
      (/ (* (* -4.5 t) z) a_m)
      (if (<= t_1 5e-88)
        (/ (* x y) (+ a_m a_m))
        (if (<= t_1 1e+294)
          (/ (* (* z t) -4.5) a_m)
          (* (* t (/ z a_m)) -4.5)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 1e+294) {
		tmp = ((z * t) * -4.5) / a_m;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-2d+19)) then
        tmp = (((-4.5d0) * t) * z) / a_m
    else if (t_1 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else if (t_1 <= 1d+294) then
        tmp = ((z * t) * (-4.5d0)) / a_m
    else
        tmp = (t * (z / a_m)) * (-4.5d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 1e+294) {
		tmp = ((z * t) * -4.5) / a_m;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -2e+19:
		tmp = ((-4.5 * t) * z) / a_m
	elif t_1 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	elif t_1 <= 1e+294:
		tmp = ((z * t) * -4.5) / a_m
	else:
		tmp = (t * (z / a_m)) * -4.5
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a_m);
	elseif (t_1 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (t_1 <= 1e+294)
		tmp = Float64(Float64(Float64(z * t) * -4.5) / a_m);
	else
		tmp = Float64(Float64(t * Float64(z / a_m)) * -4.5);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -2e+19)
		tmp = ((-4.5 * t) * z) / a_m;
	elseif (t_1 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	elseif (t_1 <= 1e+294)
		tmp = ((z * t) * -4.5) / a_m;
	else
		tmp = (t * (z / a_m)) * -4.5;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -2e+19], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+294], N[(N[(N[(z * t), $MachinePrecision] * -4.5), $MachinePrecision] / a$95$m), $MachinePrecision], N[(N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2e19

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 72.3

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

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 72.3

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

   7. Applied rewrites 72.3%

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

   if -2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.00000000000000007e294

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 64.7

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

   5. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. metadata-eval N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 64.7

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

   7. Applied rewrites 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.6

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

   9. Applied rewrites 64.6%

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

   if 1.00000000000000007e294 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 70.5

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

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. lift-/.f64 95.4

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

   7. Applied rewrites 95.4%

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

Alternative 4: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+214}:\\ \;\;\;\;\frac{t \cdot z}{a\_m} \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{z}{a\_m}\right) \cdot -4.5\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_1 -2e+19)
      (/ (* (* -4.5 t) z) a_m)
      (if (<= t_1 5e-88)
        (/ (* x y) (+ a_m a_m))
        (if (<= t_1 5e+214)
          (* (/ (* t z) a_m) -4.5)
          (* (* t (/ z a_m)) -4.5)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 5e+214) {
		tmp = ((t * z) / a_m) * -4.5;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-2d+19)) then
        tmp = (((-4.5d0) * t) * z) / a_m
    else if (t_1 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else if (t_1 <= 5d+214) then
        tmp = ((t * z) / a_m) * (-4.5d0)
    else
        tmp = (t * (z / a_m)) * (-4.5d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = ((-4.5 * t) * z) / a_m;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 5e+214) {
		tmp = ((t * z) / a_m) * -4.5;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -2e+19:
		tmp = ((-4.5 * t) * z) / a_m
	elif t_1 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	elif t_1 <= 5e+214:
		tmp = ((t * z) / a_m) * -4.5
	else:
		tmp = (t * (z / a_m)) * -4.5
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a_m);
	elseif (t_1 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (t_1 <= 5e+214)
		tmp = Float64(Float64(Float64(t * z) / a_m) * -4.5);
	else
		tmp = Float64(Float64(t * Float64(z / a_m)) * -4.5);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -2e+19)
		tmp = ((-4.5 * t) * z) / a_m;
	elseif (t_1 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	elseif (t_1 <= 5e+214)
		tmp = ((t * z) / a_m) * -4.5;
	else
		tmp = (t * (z / a_m)) * -4.5;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -2e+19], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+214], N[(N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision], N[(N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2e19

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 72.3

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

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 72.3

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

   7. Applied rewrites 72.3%

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

   if -2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999953e214

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 60.7

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

   5. Applied rewrites 60.7%

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

   if 4.99999999999999953e214 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 77.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 76.5

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

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. lift-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

Alternative 5: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\left(t \cdot z\right) \cdot \frac{-4.5}{a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+214}:\\ \;\;\;\;\frac{t \cdot z}{a\_m} \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{z}{a\_m}\right) \cdot -4.5\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_1 -2e+19)
      (* (* t z) (/ -4.5 a_m))
      (if (<= t_1 5e-88)
        (/ (* x y) (+ a_m a_m))
        (if (<= t_1 5e+214)
          (* (/ (* t z) a_m) -4.5)
          (* (* t (/ z a_m)) -4.5)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = (t * z) * (-4.5 / a_m);
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 5e+214) {
		tmp = ((t * z) / a_m) * -4.5;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-2d+19)) then
        tmp = (t * z) * ((-4.5d0) / a_m)
    else if (t_1 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else if (t_1 <= 5d+214) then
        tmp = ((t * z) / a_m) * (-4.5d0)
    else
        tmp = (t * (z / a_m)) * (-4.5d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+19) {
		tmp = (t * z) * (-4.5 / a_m);
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_1 <= 5e+214) {
		tmp = ((t * z) / a_m) * -4.5;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -2e+19:
		tmp = (t * z) * (-4.5 / a_m)
	elif t_1 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	elif t_1 <= 5e+214:
		tmp = ((t * z) / a_m) * -4.5
	else:
		tmp = (t * (z / a_m)) * -4.5
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -2e+19)
		tmp = Float64(Float64(t * z) * Float64(-4.5 / a_m));
	elseif (t_1 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (t_1 <= 5e+214)
		tmp = Float64(Float64(Float64(t * z) / a_m) * -4.5);
	else
		tmp = Float64(Float64(t * Float64(z / a_m)) * -4.5);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -2e+19)
		tmp = (t * z) * (-4.5 / a_m);
	elseif (t_1 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	elseif (t_1 <= 5e+214)
		tmp = ((t * z) / a_m) * -4.5;
	else
		tmp = (t * (z / a_m)) * -4.5;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -2e+19], N[(N[(t * z), $MachinePrecision] * N[(-4.5 / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+214], N[(N[(N[(t * z), $MachinePrecision] / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision], N[(N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2e19

   1. Initial program 87.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 72.3

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

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. metadata-eval N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 72.3

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 72.3

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

   9. Applied rewrites 72.3%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 72.3

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

   11. Applied rewrites 72.3%

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

   if -2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999953e214

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 60.7

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

   5. Applied rewrites 60.7%

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

   if 4.99999999999999953e214 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 77.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 76.5

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

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. lift-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

Alternative 6: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot \frac{-4.5}{a\_m}\\ t_2 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+214}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{z}{a\_m}\right) \cdot -4.5\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* t z) (/ -4.5 a_m))) (t_2 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_2 -2e+19)
      t_1
      (if (<= t_2 5e-88)
        (/ (* x y) (+ a_m a_m))
        (if (<= t_2 5e+214) t_1 (* (* t (/ z a_m)) -4.5)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (t * z) * (-4.5 / a_m);
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -2e+19) {
		tmp = t_1;
	} else if (t_2 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_2 <= 5e+214) {
		tmp = t_1;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) * ((-4.5d0) / a_m)
    t_2 = (z * 9.0d0) * t
    if (t_2 <= (-2d+19)) then
        tmp = t_1
    else if (t_2 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else if (t_2 <= 5d+214) then
        tmp = t_1
    else
        tmp = (t * (z / a_m)) * (-4.5d0)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (t * z) * (-4.5 / a_m);
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -2e+19) {
		tmp = t_1;
	} else if (t_2 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_2 <= 5e+214) {
		tmp = t_1;
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (t * z) * (-4.5 / a_m)
	t_2 = (z * 9.0) * t
	tmp = 0
	if t_2 <= -2e+19:
		tmp = t_1
	elif t_2 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	elif t_2 <= 5e+214:
		tmp = t_1
	else:
		tmp = (t * (z / a_m)) * -4.5
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(t * z) * Float64(-4.5 / a_m))
	t_2 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_2 <= -2e+19)
		tmp = t_1;
	elseif (t_2 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (t_2 <= 5e+214)
		tmp = t_1;
	else
		tmp = Float64(Float64(t * Float64(z / a_m)) * -4.5);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (t * z) * (-4.5 / a_m);
	t_2 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_2 <= -2e+19)
		tmp = t_1;
	elseif (t_2 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	elseif (t_2 <= 5e+214)
		tmp = t_1;
	else
		tmp = (t * (z / a_m)) * -4.5;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * N[(-4.5 / a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -2e+19], t$95$1, If[LessEqual[t$95$2, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+214], t$95$1, N[(N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2e19 or 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999953e214

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 67.0

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

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. metadata-eval N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 67.0

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

   7. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 67.0

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

   9. Applied rewrites 67.0%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 67.0

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

   11. Applied rewrites 67.0%

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

   if -2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 4.99999999999999953e214 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 77.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 76.5

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

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. lift-/.f64 92.5

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

   7. Applied rewrites 92.5%

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

Alternative 7: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot \frac{-4.5}{a\_m}\\ t_2 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+214}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* t z) (/ -4.5 a_m))) (t_2 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_2 -2e+19)
      t_1
      (if (<= t_2 5e-88)
        (/ (* x y) (+ a_m a_m))
        (if (<= t_2 5e+214) t_1 (* (/ (* -4.5 z) a_m) t)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (t * z) * (-4.5 / a_m);
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -2e+19) {
		tmp = t_1;
	} else if (t_2 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_2 <= 5e+214) {
		tmp = t_1;
	} else {
		tmp = ((-4.5 * z) / a_m) * t;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * z) * ((-4.5d0) / a_m)
    t_2 = (z * 9.0d0) * t
    if (t_2 <= (-2d+19)) then
        tmp = t_1
    else if (t_2 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else if (t_2 <= 5d+214) then
        tmp = t_1
    else
        tmp = (((-4.5d0) * z) / a_m) * t
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (t * z) * (-4.5 / a_m);
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -2e+19) {
		tmp = t_1;
	} else if (t_2 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else if (t_2 <= 5e+214) {
		tmp = t_1;
	} else {
		tmp = ((-4.5 * z) / a_m) * t;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (t * z) * (-4.5 / a_m)
	t_2 = (z * 9.0) * t
	tmp = 0
	if t_2 <= -2e+19:
		tmp = t_1
	elif t_2 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	elif t_2 <= 5e+214:
		tmp = t_1
	else:
		tmp = ((-4.5 * z) / a_m) * t
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(t * z) * Float64(-4.5 / a_m))
	t_2 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_2 <= -2e+19)
		tmp = t_1;
	elseif (t_2 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	elseif (t_2 <= 5e+214)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-4.5 * z) / a_m) * t);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (t * z) * (-4.5 / a_m);
	t_2 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_2 <= -2e+19)
		tmp = t_1;
	elseif (t_2 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	elseif (t_2 <= 5e+214)
		tmp = t_1;
	else
		tmp = ((-4.5 * z) / a_m) * t;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * N[(-4.5 / a$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -2e+19], t$95$1, If[LessEqual[t$95$2, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+214], t$95$1, N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -2e19 or 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 4.99999999999999953e214

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 67.0

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

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. metadata-eval N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 67.0

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

   7. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 67.0

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

   9. Applied rewrites 67.0%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 67.0

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

   11. Applied rewrites 67.0%

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

   if -2e19 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.9

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

   9. Applied rewrites 76.9%

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

   if 4.99999999999999953e214 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 77.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 92.4

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

   10. Applied rewrites 92.4%

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

Alternative 8: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;\left(\frac{z}{a\_m} \cdot -4.5\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a\_m} \cdot t\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_1 -2e+127)
      (* (* (/ z a_m) -4.5) t)
      (if (<= t_1 5e-88) (/ (* x y) (+ a_m a_m)) (* (/ (* -4.5 z) a_m) t))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+127) {
		tmp = ((z / a_m) * -4.5) * t;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = ((-4.5 * z) / a_m) * t;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-2d+127)) then
        tmp = ((z / a_m) * (-4.5d0)) * t
    else if (t_1 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = (((-4.5d0) * z) / a_m) * t
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -2e+127) {
		tmp = ((z / a_m) * -4.5) * t;
	} else if (t_1 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = ((-4.5 * z) / a_m) * t;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -2e+127:
		tmp = ((z / a_m) * -4.5) * t
	elif t_1 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = ((-4.5 * z) / a_m) * t
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -2e+127)
		tmp = Float64(Float64(Float64(z / a_m) * -4.5) * t);
	elseif (t_1 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(Float64(-4.5 * z) / a_m) * t);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -2e+127)
		tmp = ((z / a_m) * -4.5) * t;
	elseif (t_1 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = ((-4.5 * z) / a_m) * t;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, -2e+127], N[(N[(N[(z / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * z), $MachinePrecision] / a$95$m), $MachinePrecision] * t), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999991e127

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 87.0

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

   8. Applied rewrites 87.0%

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

   if -1.99999999999999991e127 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.8

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

   6. Applied rewrites 94.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 70.7

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

   9. Applied rewrites 70.7%

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

   if 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.5%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 66.5

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

   10. Applied rewrites 66.5%

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

Alternative 9: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_1 := \left(\frac{z}{a\_m} \cdot -4.5\right) \cdot t\\ t_2 := \left(z \cdot 9\right) \cdot t\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x \cdot y}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (let* ((t_1 (* (* (/ z a_m) -4.5) t)) (t_2 (* (* z 9.0) t)))
   (*
    a_s
    (if (<= t_2 -2e+127)
      t_1
      (if (<= t_2 5e-88) (/ (* x y) (+ a_m a_m)) t_1)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((z / a_m) * -4.5) * t;
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -2e+127) {
		tmp = t_1;
	} else if (t_2 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((z / a_m) * (-4.5d0)) * t
    t_2 = (z * 9.0d0) * t
    if (t_2 <= (-2d+127)) then
        tmp = t_1
    else if (t_2 <= 5d-88) then
        tmp = (x * y) / (a_m + a_m)
    else
        tmp = t_1
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	double t_1 = ((z / a_m) * -4.5) * t;
	double t_2 = (z * 9.0) * t;
	double tmp;
	if (t_2 <= -2e+127) {
		tmp = t_1;
	} else if (t_2 <= 5e-88) {
		tmp = (x * y) / (a_m + a_m);
	} else {
		tmp = t_1;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	t_1 = ((z / a_m) * -4.5) * t
	t_2 = (z * 9.0) * t
	tmp = 0
	if t_2 <= -2e+127:
		tmp = t_1
	elif t_2 <= 5e-88:
		tmp = (x * y) / (a_m + a_m)
	else:
		tmp = t_1
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	t_1 = Float64(Float64(Float64(z / a_m) * -4.5) * t)
	t_2 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_2 <= -2e+127)
		tmp = t_1;
	elseif (t_2 <= 5e-88)
		tmp = Float64(Float64(x * y) / Float64(a_m + a_m));
	else
		tmp = t_1;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, x, y, z, t, a_m)
	t_1 = ((z / a_m) * -4.5) * t;
	t_2 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_2 <= -2e+127)
		tmp = t_1;
	elseif (t_2 <= 5e-88)
		tmp = (x * y) / (a_m + a_m);
	else
		tmp = t_1;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := Block[{t$95$1 = N[(N[(N[(z / a$95$m), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$2, -2e+127], t$95$1, If[LessEqual[t$95$2, 5e-88], N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999991e127 or 5.00000000000000009e-88 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 87.4%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{-9}{2} \cdot \frac{z}{a} + \frac{1}{2} \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 73.6

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

   8. Applied rewrites 73.6%

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

   if -1.99999999999999991e127 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000009e-88

   1. Initial program 94.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 94.8

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

   6. Applied rewrites 94.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 70.7

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

   9. Applied rewrites 70.7%

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

Alternative 10: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 10^{+294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(z \cdot t\right) \cdot -9\right)}{a\_m + a\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \frac{z}{a\_m}\right) \cdot -4.5\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m)
 :precision binary64
 (*
  a_s
  (if (<= (* (* z 9.0) t) 1e+294)
    (/ (fma y x (* (* z t) -9.0)) (+ a_m a_m))
    (* (* t (/ z a_m)) -4.5))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	double tmp;
	if (((z * 9.0) * t) <= 1e+294) {
		tmp = fma(y, x, ((z * t) * -9.0)) / (a_m + a_m);
	} else {
		tmp = (t * (z / a_m)) * -4.5;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= 1e+294)
		tmp = Float64(fma(y, x, Float64(Float64(z * t) * -9.0)) / Float64(a_m + a_m));
	else
		tmp = Float64(Float64(t * Float64(z / a_m)) * -4.5);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], 1e+294], N[(N[(y * x + N[(N[(z * t), $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(z / a$95$m), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.00000000000000007e294

   1. Initial program 93.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. 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. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 93.1

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

   6. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if 1.00000000000000007e294 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. 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 70.5

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

   5. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
   6. 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. lift-/.f64 95.4

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

   7. Applied rewrites 95.4%

      \[\leadsto \left(t \cdot \frac{z}{a}\right) \cdot -4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 49.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \frac{x \cdot y}{a\_m + a\_m} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s x y z t a_m) :precision binary64 (* a_s (/ (* x y) (+ a_m a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / (a_m + a_m));
}

Fortran

a\_m =     private
a\_s =     private
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(a_s, x, y, z, t, a_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    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_m
    code = a_s * ((x * y) / (a_m + a_m))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double x, double y, double z, double t, double a_m) {
	return a_s * ((x * y) / (a_m + a_m));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, x, y, z, t, a_m):
	return a_s * ((x * y) / (a_m + a_m))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, x, y, z, t, a_m)
	return Float64(a_s * Float64(Float64(x * y) / Float64(a_m + a_m)))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, x, y, z, t, a_m)
	tmp = a_s * ((x * y) / (a_m + a_m));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, x_, y_, z_, t_, a$95$m_] := N[(a$95$s * N[(N[(x * y), $MachinePrecision] / N[(a$95$m + a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Add Preprocessing
3. 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. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 91.5

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

4. Applied rewrites 91.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 91.5

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

6. Applied rewrites 91.5%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 49.6

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

9. Applied rewrites 49.6%

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

Developer Target 1: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
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) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025091 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))