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

Percentage Accurate: 91.4% → 96.7%

Time: 9.1s

Alternatives: 7

Speedup: 0.7×

• 28.0% 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

real(8) function code(x, y, z, t, a)
    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: 91.4% 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

real(8) function code(x, y, z, t, a)
    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: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 0.5, -4.5 \cdot t\right)}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a) x)
     (if (<= t_1 2e+305)
       (/ (fma y x (* (* -9.0 z) t)) (+ a a))
       (* (/ (fma (* (/ x z) y) 0.5 (* -4.5 t)) a) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a) * x;
	} else if (t_1 <= 2e+305) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	} else {
		tmp = (fma(((x / z) * y), 0.5, (-4.5 * t)) / a) * z;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a) * x);
	elseif (t_1 <= 2e+305)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x / z) * y), 0.5, Float64(-4.5 * t)) / a) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+305], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] * 0.5 + N[(-4.5 * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 68.9%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 90.6%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.9999999999999999e305

   1. Initial program 99.1%

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

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

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

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

      4. lift-*.f64 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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.1

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

   6. Applied rewrites 99.1%

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

   if 1.9999999999999999e305 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 61.7%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. distribute-lft-neg-out N/A

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{z} \cdot y, 0.5, -4.5 \cdot t\right)}{a} \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, y, t \cdot \left(\frac{z}{x} \cdot -4.5\right)\right)}{a} \cdot x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot y, \frac{0.5}{z}, -4.5 \cdot t\right)}{a} \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))))
   (if (<= t_1 (- INFINITY))
     (* (/ (fma 0.5 y (* t (* (/ z x) -4.5))) a) x)
     (if (<= t_1 5e+265)
       (/ (fma y x (* (* -9.0 z) t)) (+ a a))
       (* (/ (fma (* x y) (/ 0.5 z) (* -4.5 t)) a) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(0.5, y, (t * ((z / x) * -4.5))) / a) * x;
	} else if (t_1 <= 5e+265) {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	} else {
		tmp = (fma((x * y), (0.5 / z), (-4.5 * t)) / a) * z;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(0.5, y, Float64(t * Float64(Float64(z / x) * -4.5))) / a) * x);
	elseif (t_1 <= 5e+265)
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	else
		tmp = Float64(Float64(fma(Float64(x * y), Float64(0.5 / z), Float64(-4.5 * t)) / a) * z);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * y + N[(t * N[(N[(z / x), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 5e+265], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * y), $MachinePrecision] * N[(0.5 / z), $MachinePrecision] + N[(-4.5 * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 74.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out-- N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64))) < 5.0000000000000002e265

   1. Initial program 98.9%

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

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

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

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

      4. lift-*.f64 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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if 5.0000000000000002e265 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) (*.f64 a #s(literal 2 binary64)))

   1. Initial program 82.9%

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

   3. Taylor expanded in z around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. distribute-lft-neg-out N/A

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

      8. *-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 85.5%

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot y, \frac{0.5}{z}, -4.5 \cdot t\right)}{a} \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+146} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (or (<= t_1 -2e+146) (not (<= t_1 2e-23)))
     (* (/ (* -4.5 t) a) z)
     (/ (* x y) (+ a a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23)) {
		tmp = ((-4.5 * t) / a) * z;
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if ((t_1 <= (-2d+146)) .or. (.not. (t_1 <= 2d-23))) then
        tmp = (((-4.5d0) * t) / a) * z
    else
        tmp = (x * y) / (a + a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23)) {
		tmp = ((-4.5 * t) / a) * z;
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if (t_1 <= -2e+146) or not (t_1 <= 2e-23):
		tmp = ((-4.5 * t) / a) * z
	else:
		tmp = (x * y) / (a + a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23))
		tmp = Float64(Float64(Float64(-4.5 * t) / a) * z);
	else
		tmp = Float64(Float64(x * y) / Float64(a + a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if ((t_1 <= -2e+146) || ~((t_1 <= 2e-23)))
		tmp = ((-4.5 * t) / a) * z;
	else
		tmp = (x * y) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+146], N[Not[LessEqual[t$95$1, 2e-23]], $MachinePrecision]], N[(N[(N[(-4.5 * t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999987e146 or 1.99999999999999992e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 85.0%

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

   if -1.99999999999999987e146 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.99999999999999992e-23

   1. Initial program 96.4%

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

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

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

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

      4. lift-*.f64 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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 77.1

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

   9. Applied rewrites 77.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+146} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+146} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;z \cdot \left(\frac{t}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (or (<= t_1 -2e+146) (not (<= t_1 2e-23)))
     (* z (* (/ t a) -4.5))
     (/ (* x y) (+ a a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23)) {
		tmp = z * ((t / a) * -4.5);
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if ((t_1 <= (-2d+146)) .or. (.not. (t_1 <= 2d-23))) then
        tmp = z * ((t / a) * (-4.5d0))
    else
        tmp = (x * y) / (a + a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23)) {
		tmp = z * ((t / a) * -4.5);
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if (t_1 <= -2e+146) or not (t_1 <= 2e-23):
		tmp = z * ((t / a) * -4.5)
	else:
		tmp = (x * y) / (a + a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23))
		tmp = Float64(z * Float64(Float64(t / a) * -4.5));
	else
		tmp = Float64(Float64(x * y) / Float64(a + a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if ((t_1 <= -2e+146) || ~((t_1 <= 2e-23)))
		tmp = z * ((t / a) * -4.5);
	else
		tmp = (x * y) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+146], N[Not[LessEqual[t$95$1, 2e-23]], $MachinePrecision]], N[(z * N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999987e146 or 1.99999999999999992e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 85.0%

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

   if -1.99999999999999987e146 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.99999999999999992e-23

   1. Initial program 96.4%

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

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

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

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

      4. lift-*.f64 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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 77.1

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

   9. Applied rewrites 77.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+146} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;z \cdot \left(\frac{t}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+146} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (or (<= t_1 -2e+146) (not (<= t_1 2e-23)))
     (* t (* z (/ -4.5 a)))
     (/ (* x y) (+ a a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if ((t_1 <= (-2d+146)) .or. (.not. (t_1 <= 2d-23))) then
        tmp = t * (z * ((-4.5d0) / a))
    else
        tmp = (x * y) / (a + a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23)) {
		tmp = t * (z * (-4.5 / a));
	} else {
		tmp = (x * y) / (a + a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if (t_1 <= -2e+146) or not (t_1 <= 2e-23):
		tmp = t * (z * (-4.5 / a))
	else:
		tmp = (x * y) / (a + a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if ((t_1 <= -2e+146) || !(t_1 <= 2e-23))
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	else
		tmp = Float64(Float64(x * y) / Float64(a + a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if ((t_1 <= -2e+146) || ~((t_1 <= 2e-23)))
		tmp = t * (z * (-4.5 / a));
	else
		tmp = (x * y) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+146], N[Not[LessEqual[t$95$1, 2e-23]], $MachinePrecision]], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999987e146 or 1.99999999999999992e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 86.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.9%

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

   if -1.99999999999999987e146 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.99999999999999992e-23

   1. Initial program 96.4%

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

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

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

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

      4. lift-*.f64 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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 77.1

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

   9. Applied rewrites 77.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+146} \lor \neg \left(\left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{-23}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -5 \cdot 10^{+253}:\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(-9 \cdot z\right) \cdot t\right)}{a + a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) -5e+253)
   (* (/ (* -4.5 t) a) z)
   (/ (fma y x (* (* -9.0 z) t)) (+ a a))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= -5e+253) {
		tmp = ((-4.5 * t) / a) * z;
	} else {
		tmp = fma(y, x, ((-9.0 * z) * t)) / (a + a);
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= -5e+253)
		tmp = Float64(Float64(Float64(-4.5 * t) / a) * z);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-9.0 * z) * t)) / Float64(a + a));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], -5e+253], N[(N[(N[(-4.5 * t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision], N[(N[(y * x + N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.9999999999999997e253

   1. Initial program 63.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.6%

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

   if -4.9999999999999997e253 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 95.2%

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

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

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

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

      4. lift-*.f64 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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 95.2

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

   6. Applied rewrites 95.2%

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

Alternative 7: 51.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x * y) / (a + a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

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

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

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

   4. lift-*.f64 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

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 92.4

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

4. Applied rewrites 92.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 92.4

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

6. Applied rewrites 92.4%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 51.7

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

9. Applied rewrites 51.7%

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

Developer Target 1: 93.0% 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

real(8) function code(x, y, z, t, a)
    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 2024337 
(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)))