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

Percentage Accurate: 90.9% → 90.9%

Time: 6.6s

Alternatives: 6

Speedup: 1.1×

• 27.6% 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: 90.9% 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: 90.9% accurate, 1.0× speedup

Math

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

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) (* (* z 9.0) t)) (* a 2.0)))

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) - ((z * 9.0) * t)) / (a * 2.0);
}

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) - ((z * 9.0d0) * t)) / (a * 2.0d0)
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) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

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

Julia

x, y, z, t, a = sort([x, y, z, t, a])
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

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
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[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

Alternative 2: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a \cdot 2}\\ \mathbf{if}\;x \cdot y \leq -2.15 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x y) (* a 2.0))))
   (if (<= (* x y) -2.15e+63)
     t_1
     (if (<= (* x y) 3.6e+15) (/ (* t (* z -9.0)) (* a 2.0)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / (a * 2.0);
	double tmp;
	if ((x * y) <= -2.15e+63) {
		tmp = t_1;
	} else if ((x * y) <= 3.6e+15) {
		tmp = (t * (z * -9.0)) / (a * 2.0);
	} else {
		tmp = t_1;
	}
	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 = (x * y) / (a * 2.0d0)
    if ((x * y) <= (-2.15d+63)) then
        tmp = t_1
    else if ((x * y) <= 3.6d+15) then
        tmp = (t * (z * (-9.0d0))) / (a * 2.0d0)
    else
        tmp = t_1
    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 = (x * y) / (a * 2.0);
	double tmp;
	if ((x * y) <= -2.15e+63) {
		tmp = t_1;
	} else if ((x * y) <= 3.6e+15) {
		tmp = (t * (z * -9.0)) / (a * 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) / (a * 2.0)
	tmp = 0
	if (x * y) <= -2.15e+63:
		tmp = t_1
	elif (x * y) <= 3.6e+15:
		tmp = (t * (z * -9.0)) / (a * 2.0)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / Float64(a * 2.0))
	tmp = 0.0
	if (Float64(x * y) <= -2.15e+63)
		tmp = t_1;
	elseif (Float64(x * y) <= 3.6e+15)
		tmp = Float64(Float64(t * Float64(z * -9.0)) / Float64(a * 2.0));
	else
		tmp = t_1;
	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 = (x * y) / (a * 2.0);
	tmp = 0.0;
	if ((x * y) <= -2.15e+63)
		tmp = t_1;
	elseif ((x * y) <= 3.6e+15)
		tmp = (t * (z * -9.0)) / (a * 2.0);
	else
		tmp = t_1;
	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[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.15e+63], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 3.6e+15], N[(N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.15e63 or 3.6e15 < (*.f64 x y)

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. lower-*.f64 81.7

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   5. Simplified 81.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   if -2.15e63 < (*.f64 x y) < 3.6e15

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.4

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

   8. Simplified 78.4%

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

Alternative 3: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{x \cdot y}{a \cdot 2}\\ \mathbf{if}\;x \cdot y \leq -2.15 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x y) (* a 2.0))))
   (if (<= (* x y) -2.15e+63)
     t_1
     (if (<= (* x y) 3.6e+15) (/ (* t (* z -4.5)) a) t_1))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) / (a * 2.0);
	double tmp;
	if ((x * y) <= -2.15e+63) {
		tmp = t_1;
	} else if ((x * y) <= 3.6e+15) {
		tmp = (t * (z * -4.5)) / a;
	} else {
		tmp = t_1;
	}
	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 = (x * y) / (a * 2.0d0)
    if ((x * y) <= (-2.15d+63)) then
        tmp = t_1
    else if ((x * y) <= 3.6d+15) then
        tmp = (t * (z * (-4.5d0))) / a
    else
        tmp = t_1
    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 = (x * y) / (a * 2.0);
	double tmp;
	if ((x * y) <= -2.15e+63) {
		tmp = t_1;
	} else if ((x * y) <= 3.6e+15) {
		tmp = (t * (z * -4.5)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) / (a * 2.0)
	tmp = 0
	if (x * y) <= -2.15e+63:
		tmp = t_1
	elif (x * y) <= 3.6e+15:
		tmp = (t * (z * -4.5)) / a
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) / Float64(a * 2.0))
	tmp = 0.0
	if (Float64(x * y) <= -2.15e+63)
		tmp = t_1;
	elseif (Float64(x * y) <= 3.6e+15)
		tmp = Float64(Float64(t * Float64(z * -4.5)) / a);
	else
		tmp = t_1;
	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 = (x * y) / (a * 2.0);
	tmp = 0.0;
	if ((x * y) <= -2.15e+63)
		tmp = t_1;
	elseif ((x * y) <= 3.6e+15)
		tmp = (t * (z * -4.5)) / a;
	else
		tmp = t_1;
	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[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.15e+63], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 3.6e+15], N[(N[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.15e63 or 3.6e15 < (*.f64 x y)

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]
   4. Step-by-step derivation

      1. lower-*.f64 81.7

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   5. Simplified 81.7%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} \]

   if -2.15e63 < (*.f64 x y) < 3.6e15

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 94.9

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

   5. Simplified 94.9%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.4

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

   8. Simplified 78.4%

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

   9. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 78.4

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

   11. Simplified 78.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.15 \cdot 10^{+63}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 1.1× speedup

Math

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

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

C

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

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(fma(Float64(z * t), -9.0, Float64(x * y)) / Float64(a * 2.0))
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[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Final simplification 92.6%

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

Alternative 5: 50.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{t \cdot \left(z \cdot -4.5\right)}{a} \end{array} \]

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 (/ (* t (* z -4.5)) a))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (t * (z * -4.5)) / 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 = (t * (z * (-4.5d0))) / 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 (t * (z * -4.5)) / a;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (t * (z * -4.5)) / a

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(t * Float64(z * -4.5)) / a)
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (t * (z * -4.5)) / 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[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in x around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 92.6

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

5. Simplified 92.6%

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

6. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 55.9

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

8. Simplified 55.9%

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

9. Taylor expanded in t around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 55.9

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

11. Simplified 55.9%

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

12. Final simplification 55.9%

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

Alternative 6: 50.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \frac{\left(z \cdot t\right) \cdot -4.5}{a} \end{array} \]

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 (/ (* (* z t) -4.5) a))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return ((z * t) * -4.5) / 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 = ((z * t) * (-4.5d0)) / 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 ((z * t) * -4.5) / a;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return ((z * t) * -4.5) / a

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(Float64(z * t) * -4.5) / a)
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = ((z * t) * -4.5) / 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[(N[(z * t), $MachinePrecision] * -4.5), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 55.9

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

5. Simplified 55.9%

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

6. Final simplification 55.9%

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

Developer Target 1: 93.3% 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 2024215 
(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)))