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

Percentage Accurate: 95.8% → 96.9%

Time: 9.5s

Alternatives: 14

Speedup: 0.5×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{t}{\left(z \cdot y\right) \cdot 3} + \left(x - \frac{\frac{y}{3}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (/ t (* (* 3.0 z) y)) (- x (/ y (* 3.0 z)))) 2e+227)
   (+ (/ t (* (* z y) 3.0)) (- x (/ (/ y 3.0) z)))
   (- x (/ (- y (/ t y)) (* 3.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))) <= 2e+227) {
		tmp = (t / ((z * y) * 3.0)) + (x - ((y / 3.0) / z));
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t / ((3.0d0 * z) * y)) + (x - (y / (3.0d0 * z)))) <= 2d+227) then
        tmp = (t / ((z * y) * 3.0d0)) + (x - ((y / 3.0d0) / z))
    else
        tmp = x - ((y - (t / y)) / (3.0d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))) <= 2e+227) {
		tmp = (t / ((z * y) * 3.0)) + (x - ((y / 3.0) / z));
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))) <= 2e+227:
		tmp = (t / ((z * y) * 3.0)) + (x - ((y / 3.0) / z))
	else:
		tmp = x - ((y - (t / y)) / (3.0 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + Float64(x - Float64(y / Float64(3.0 * z)))) <= 2e+227)
		tmp = Float64(Float64(t / Float64(Float64(z * y) * 3.0)) + Float64(x - Float64(Float64(y / 3.0) / z)));
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))) <= 2e+227)
		tmp = (t / ((z * y) * 3.0)) + (x - ((y / 3.0) / z));
	else
		tmp = x - ((y - (t / y)) / (3.0 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+227], N[(N[(t / N[(N[(z * y), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(y / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.0000000000000002e227

   1. Initial program 98.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right)} \cdot y} \]

      3. associate-*l* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right) \cdot 3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right) \cdot 3}} \]

      7. lower-*.f64 98.8

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right)} \cdot 3} \]

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{\color{blue}{z \cdot 3}}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(x - \color{blue}{\frac{y}{z \cdot 3}}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{\color{blue}{3 \cdot z}}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

      4. associate-/r* N/A

         \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{3}}{z}}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{3}}{z}}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

      6. lower-/.f64 98.8

         \[\leadsto \left(x - \frac{\color{blue}{\frac{y}{3}}}{z}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

   6. Applied rewrites 98.8%

      \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{3}}{z}}\right) + \frac{t}{\left(z \cdot y\right) \cdot 3} \]

   if 2.0000000000000002e227 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 86.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.9

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.9

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{t}{\left(z \cdot y\right) \cdot 3} + \left(x - \frac{\frac{y}{3}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{3 \cdot z}\\ \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + t\_1 \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{t}{\left(z \cdot y\right) \cdot 3} + t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* 3.0 z)))))
   (if (<= (+ (/ t (* (* 3.0 z) y)) t_1) 2e+227)
     (+ (/ t (* (* z y) 3.0)) t_1)
     (- x (/ (- y (/ t y)) (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (((t / ((3.0 * z) * y)) + t_1) <= 2e+227) {
		tmp = (t / ((z * y) * 3.0)) + t_1;
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (3.0d0 * z))
    if (((t / ((3.0d0 * z) * y)) + t_1) <= 2d+227) then
        tmp = (t / ((z * y) * 3.0d0)) + t_1
    else
        tmp = x - ((y - (t / y)) / (3.0d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (((t / ((3.0 * z) * y)) + t_1) <= 2e+227) {
		tmp = (t / ((z * y) * 3.0)) + t_1;
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / (3.0 * z))
	tmp = 0
	if ((t / ((3.0 * z) * y)) + t_1) <= 2e+227:
		tmp = (t / ((z * y) * 3.0)) + t_1
	else:
		tmp = x - ((y - (t / y)) / (3.0 * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(3.0 * z)))
	tmp = 0.0
	if (Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + t_1) <= 2e+227)
		tmp = Float64(Float64(t / Float64(Float64(z * y) * 3.0)) + t_1);
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (3.0 * z));
	tmp = 0.0;
	if (((t / ((3.0 * z) * y)) + t_1) <= 2e+227)
		tmp = (t / ((z * y) * 3.0)) + t_1;
	else
		tmp = x - ((y - (t / y)) / (3.0 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], 2e+227], N[(N[(t / N[(N[(z * y), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.0000000000000002e227

   1. Initial program 98.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot 3\right)} \cdot y} \]

      3. associate-*l* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right) \cdot 3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right) \cdot 3}} \]

      7. lower-*.f64 98.8

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right)} \cdot 3} \]

   4. Applied rewrites 98.8%

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right) \cdot 3}} \]

   if 2.0000000000000002e227 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 86.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.9

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.9

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{t}{\left(z \cdot y\right) \cdot 3} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ t (* (* 3.0 z) y)) (- x (/ y (* 3.0 z))))))
   (if (<= t_1 2e+227) t_1 (- x (/ (- y (/ t y)) (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	double tmp;
	if (t_1 <= 2e+227) {
		tmp = t_1;
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / ((3.0d0 * z) * y)) + (x - (y / (3.0d0 * z)))
    if (t_1 <= 2d+227) then
        tmp = t_1
    else
        tmp = x - ((y - (t / y)) / (3.0d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	double tmp;
	if (t_1 <= 2e+227) {
		tmp = t_1;
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))
	tmp = 0
	if t_1 <= 2e+227:
		tmp = t_1
	else:
		tmp = x - ((y - (t / y)) / (3.0 * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + Float64(x - Float64(y / Float64(3.0 * z))))
	tmp = 0.0
	if (t_1 <= 2e+227)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	tmp = 0.0;
	if (t_1 <= 2e+227)
		tmp = t_1;
	else
		tmp = x - ((y - (t / y)) / (3.0 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+227], t$95$1, N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.0000000000000002e227

   1. Initial program 98.8%

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

   if 2.0000000000000002e227 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 86.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.9

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.9

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z \cdot y}, t, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (/ t (* (* 3.0 z) y)) (- x (/ y (* 3.0 z)))) 2e+227)
   (fma (/ 0.3333333333333333 (* z y)) t (fma (/ y z) -0.3333333333333333 x))
   (- x (/ (- y (/ t y)) (* 3.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))) <= 2e+227) {
		tmp = fma((0.3333333333333333 / (z * y)), t, fma((y / z), -0.3333333333333333, x));
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + Float64(x - Float64(y / Float64(3.0 * z)))) <= 2e+227)
		tmp = fma(Float64(0.3333333333333333 / Float64(z * y)), t, fma(Float64(y / z), -0.3333333333333333, x));
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+227], N[(N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] * t + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.0000000000000002e227

   1. Initial program 98.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 95.8

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 95.8

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 95.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

      3. lift--.f64 N/A

         \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{3 \cdot z} \]

      4. div-sub N/A

         \[\leadsto x - \color{blue}{\left(\frac{y}{3 \cdot z} - \frac{\frac{t}{y}}{3 \cdot z}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{\color{blue}{3 \cdot z}} - \frac{\frac{t}{y}}{3 \cdot z}\right) \]

      6. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{\color{blue}{z \cdot 3}} - \frac{\frac{t}{y}}{3 \cdot z}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{\frac{t}{y}}{3 \cdot z}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{\color{blue}{z \cdot 3}} - \frac{\frac{t}{y}}{3 \cdot z}\right) \]

      9. associate--r- N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{3 \cdot z}} \]

      10. lift--.f64 N/A

          \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{\frac{t}{y}}{3 \cdot z} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{t}{y}}}{3 \cdot z} \]

      13. *-commutative N/A

          \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      14. associate-/l/ N/A

          \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      15. +-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y} + \left(x - \frac{y}{z \cdot 3}\right)} \]

      16. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(z \cdot 3\right) \cdot y}{t}}} + \left(x - \frac{y}{z \cdot 3}\right) \]

      17. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{\left(z \cdot 3\right) \cdot y} \cdot t} + \left(x - \frac{y}{z \cdot 3}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\left(z \cdot 3\right) \cdot y}, t, x - \frac{y}{z \cdot 3}\right)} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{z \cdot y}, t, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)} \]

   if 2.0000000000000002e227 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 86.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

      8. *-commutative N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

      9. associate-/r* N/A

         \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

      10. sub-div N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      11. lower-/.f64 N/A

          \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

      12. lower--.f64 N/A

          \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

      13. lower-/.f64 99.9

          \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

      15. *-commutative N/A

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

      16. lower-*.f64 99.9

          \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   4. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z \cdot y}, t, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{\frac{y - \frac{t}{y}}{z}}{3} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (/ (- y (/ t y)) z) 3.0)))

C

double code(double x, double y, double z, double t) {
	return x - (((y - (t / y)) / z) / 3.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y - (t / y)) / z) / 3.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y - (t / y)) / z) / 3.0);
}

Python

def code(x, y, z, t):
	return x - (((y - (t / y)) / z) / 3.0)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y - (t / y)) / z) / 3.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

   8. *-commutative N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   9. associate-/r* N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

   10. sub-div N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   11. lower-/.f64 N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   12. lower--.f64 N/A

       \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

   13. lower-/.f64 96.8

       \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

   14. lift-*.f64 N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   15. *-commutative N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   16. lower-*.f64 96.8

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

4. Applied rewrites 96.8%

   \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

   2. lift-*.f64 N/A

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   3. *-commutative N/A

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. associate-/r* N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{z}}}{3} \]

   6. lower-/.f64 96.9

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

6. Applied rewrites 96.9%

   \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]
7. Add Preprocessing

Alternative 6: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{\frac{y - \frac{t}{y}}{3}}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (/ (- y (/ t y)) 3.0) z)))

C

double code(double x, double y, double z, double t) {
	return x - (((y - (t / y)) / 3.0) / z);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y - (t / y)) / 3.0d0) / z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (((y - (t / y)) / 3.0) / z);
}

Python

def code(x, y, z, t):
	return x - (((y - (t / y)) / 3.0) / z)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / 3.0) / z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y - (t / y)) / 3.0) / z);
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

   8. *-commutative N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   9. associate-/r* N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

   10. sub-div N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   11. lower-/.f64 N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   12. lower--.f64 N/A

       \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

   13. lower-/.f64 96.8

       \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

   14. lift-*.f64 N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   15. *-commutative N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   16. lower-*.f64 96.8

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

4. Applied rewrites 96.8%

   \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

   2. lift-*.f64 N/A

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   3. *-commutative N/A

      \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   4. associate-/r* N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{z}}}{3} \]

   6. lower-/.f64 96.9

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

6. Applied rewrites 96.9%

   \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{z}}{3}} \]

   2. lift-/.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{z}}}{3} \]

   3. associate-/l/ N/A

      \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{3 \cdot z}} \]

   4. associate-/r* N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   5. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]

   6. lower-/.f64 96.8

      \[\leadsto x - \frac{\color{blue}{\frac{y - \frac{t}{y}}{3}}}{z} \]

8. Applied rewrites 96.8%

   \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]
9. Add Preprocessing

Alternative 7: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.95 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -1.3e+62)
     t_1
     (if (<= y 3.95e+28) (fma (/ t (* z y)) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -1.3e+62) {
		tmp = t_1;
	} else if (y <= 3.95e+28) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -1.3e+62)
		tmp = t_1;
	elseif (y <= 3.95e+28)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.3e+62], t$95$1, If[LessEqual[y, 3.95e+28], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999992e62 or 3.9499999999999998e28 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{y}{z}} \]

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{\frac{-1}{3}} \cdot \frac{y}{z} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, x\right)} \]

      5. lower-/.f64 98.8

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.29999999999999992e62 < y < 3.9499999999999998e28

   1. Initial program 95.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. sub-neg N/A

         \[\leadsto x - \color{blue}{\left(\frac{y}{z \cdot 3} + \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right)} \]

      5. associate--r+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) - \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} - \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) - \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) - \frac{\frac{t}{-3 \cdot z}}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot y - \frac{-1}{3} \cdot \frac{t}{z}}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{y} - \frac{\frac{-1}{3} \cdot \frac{t}{z}}{y}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{y}} - \frac{\frac{-1}{3} \cdot \frac{t}{z}}{y} \]

      3. *-inverses N/A

         \[\leadsto x \cdot \color{blue}{1} - \frac{\frac{-1}{3} \cdot \frac{t}{z}}{y} \]

      4. *-rgt-identity N/A

         \[\leadsto \color{blue}{x} - \frac{\frac{-1}{3} \cdot \frac{t}{z}}{y} \]

      5. associate-/l* N/A

         \[\leadsto x - \color{blue}{\frac{-1}{3} \cdot \frac{\frac{t}{z}}{y}} \]

      6. associate-/l/ N/A

         \[\leadsto x - \frac{-1}{3} \cdot \color{blue}{\frac{t}{y \cdot z}} \]

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

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \frac{t}{y \cdot z}} \]

      8. metadata-eval N/A

         \[\leadsto x + \color{blue}{\frac{1}{3}} \cdot \frac{t}{y \cdot z} \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z} + x} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} + x \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{y \cdot z}, \frac{1}{3}, x\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{y \cdot z}}, \frac{1}{3}, x\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, \frac{1}{3}, x\right) \]

      14. lower-*.f64 89.6

          \[\leadsto \mathsf{fma}\left(\frac{t}{\color{blue}{z \cdot y}}, 0.3333333333333333, x\right) \]

   7. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-106}:\\ \;\;\;\;\frac{t}{\left(3 \cdot y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -1.65e+25) t_1 (if (<= y 2e-106) (/ t (* (* 3.0 y) z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -1.65e+25) {
		tmp = t_1;
	} else if (y <= 2e-106) {
		tmp = t / ((3.0 * y) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -1.65e+25)
		tmp = t_1;
	elseif (y <= 2e-106)
		tmp = Float64(t / Float64(Float64(3.0 * y) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.65e+25], t$95$1, If[LessEqual[y, 2e-106], N[(t / N[(N[(3.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6500000000000001e25 or 1.99999999999999988e-106 < y

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{y}{z}} \]

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{\frac{-1}{3}} \cdot \frac{y}{z} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, x\right)} \]

      5. lower-/.f64 89.1

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.6500000000000001e25 < y < 1.99999999999999988e-106

   1. Initial program 94.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

      4. sub-neg N/A

         \[\leadsto x - \color{blue}{\left(\frac{y}{z \cdot 3} + \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right)} \]

      5. associate--r+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) - \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} - \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) - \left(\mathsf{neg}\left(\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) - \frac{\frac{t}{-3 \cdot z}}{y}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z}} \cdot \frac{1}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot \frac{1}{3} \]

      5. lower-*.f64 62.3

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot 0.3333333333333333 \]

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{t}{z \cdot y} \cdot 0.3333333333333333} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.3%

      \[\leadsto \frac{t}{\color{blue}{\left(3 \cdot z\right) \cdot y}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.3%

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

Alternative 9: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-106}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -1.65e+25)
     t_1
     (if (<= y 2e-106) (* (/ t (* z y)) 0.3333333333333333) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -1.65e+25) {
		tmp = t_1;
	} else if (y <= 2e-106) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -1.65e+25)
		tmp = t_1;
	elseif (y <= 2e-106)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.65e+25], t$95$1, If[LessEqual[y, 2e-106], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6500000000000001e25 or 1.99999999999999988e-106 < y

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x - \frac{1}{3} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{y}{z}} \]

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{\frac{-1}{3}} \cdot \frac{y}{z} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, x\right)} \]

      5. lower-/.f64 89.1

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]

   if -1.6500000000000001e25 < y < 1.99999999999999988e-106

   1. Initial program 94.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z} \cdot \frac{1}{3}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t}{y \cdot z}} \cdot \frac{1}{3} \]

      4. *-commutative N/A

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot \frac{1}{3} \]

      5. lower-*.f64 62.3

         \[\leadsto \frac{t}{\color{blue}{z \cdot y}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{t}{z \cdot y} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - \frac{t}{y}}{3 \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (- y (/ t y)) (* 3.0 z))))

C

double code(double x, double y, double z, double t) {
	return x - ((y - (t / y)) / (3.0 * z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - ((y - (t / y)) / (3.0d0 * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - ((y - (t / y)) / (3.0 * z));
}

Python

def code(x, y, z, t):
	return x - ((y - (t / y)) / (3.0 * z))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - ((y - (t / y)) / (3.0 * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x - \frac{y}{z \cdot 3}\right)} + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto x - \left(\color{blue}{\frac{y}{z \cdot 3}} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{\left(z \cdot 3\right) \cdot y}}\right) \]

   8. *-commutative N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   9. associate-/r* N/A

      \[\leadsto x - \left(\frac{y}{z \cdot 3} - \color{blue}{\frac{\frac{t}{y}}{z \cdot 3}}\right) \]

   10. sub-div N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   11. lower-/.f64 N/A

       \[\leadsto x - \color{blue}{\frac{y - \frac{t}{y}}{z \cdot 3}} \]

   12. lower--.f64 N/A

       \[\leadsto x - \frac{\color{blue}{y - \frac{t}{y}}}{z \cdot 3} \]

   13. lower-/.f64 96.8

       \[\leadsto x - \frac{y - \color{blue}{\frac{t}{y}}}{z \cdot 3} \]

   14. lift-*.f64 N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{z \cdot 3}} \]

   15. *-commutative N/A

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

   16. lower-*.f64 96.8

       \[\leadsto x - \frac{y - \frac{t}{y}}{\color{blue}{3 \cdot z}} \]

4. Applied rewrites 96.8%

   \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{3 \cdot z}} \]
5. Add Preprocessing

Alternative 11: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ (- y (/ t y)) z) -0.3333333333333333 x))

C

double code(double x, double y, double z, double t) {
	return fma(((y - (t / y)) / z), -0.3333333333333333, x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(Float64(y - Float64(t / y)) / z), -0.3333333333333333, x)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in t around 0

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

   1. associate--l+ N/A

      \[\leadsto \color{blue}{x + \left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} - \frac{1}{3} \cdot \frac{y}{z}\right) + x} \]

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

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(\frac{t}{y \cdot z} - \frac{y}{z}\right)} + x \]

   4. associate-/r* N/A

      \[\leadsto \frac{1}{3} \cdot \left(\color{blue}{\frac{\frac{t}{y}}{z}} - \frac{y}{z}\right) + x \]

   5. div-sub N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{\frac{t}{y} - y}{z}} + x \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(\frac{t}{y} - y\right)}{z}} + x \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{-1}{3}\right)} \cdot \left(\frac{t}{y} - y\right)}{z} + x \]

   8. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\frac{-1}{3} \cdot \left(\frac{t}{y} - y\right)\right)}}{z} + x \]

   9. distribute-lft-out-- N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1}{3} \cdot \frac{t}{y} - \frac{-1}{3} \cdot y\right)}}{z} + x \]

   10. associate-*r/ N/A

       \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{3} \cdot \frac{t}{y} - \frac{-1}{3} \cdot y}{z}} + x \]

5. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right)} \]
6. Add Preprocessing

Alternative 12: 64.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma -0.3333333333333333 (/ y z) x))

C

double code(double x, double y, double z, double t) {
	return fma(-0.3333333333333333, (y / z), x);
}

Julia

function code(x, y, z, t)
	return fma(-0.3333333333333333, Float64(y / z), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{x - \frac{1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{y}{z}} \]

   2. metadata-eval N/A

      \[\leadsto x + \color{blue}{\frac{-1}{3}} \cdot \frac{y}{z} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z} + x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, x\right)} \]

   5. lower-/.f64 65.2

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{y}{z}}, x\right) \]

5. Applied rewrites 65.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)} \]
6. Add Preprocessing

Alternative 13: 36.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{y}{-3 \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ y (* -3.0 z)))

C

double code(double x, double y, double z, double t) {
	return y / (-3.0 * z);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y / ((-3.0d0) * z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return y / (-3.0 * z);
}

Python

def code(x, y, z, t):
	return y / (-3.0 * z)

Julia

function code(x, y, z, t)
	return Float64(y / Float64(-3.0 * z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y / (-3.0 * z);
end

Wolfram

code[x_, y_, z_, t_] := N[(y / N[(-3.0 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]

   2. lower-/.f64 34.9

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]

5. Applied rewrites 34.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y}{z}} \]
6. Step-by-step derivation

7. Applied rewrites 34.9%

   \[\leadsto \frac{y}{\color{blue}{-3 \cdot z}} \]
8. Add Preprocessing

Alternative 14: 36.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{y}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* -0.3333333333333333 (/ y z)))

C

double code(double x, double y, double z, double t) {
	return -0.3333333333333333 * (y / z);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-0.3333333333333333d0) * (y / z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return -0.3333333333333333 * (y / z);
}

Python

def code(x, y, z, t):
	return -0.3333333333333333 * (y / z)

Julia

function code(x, y, z, t)
	return Float64(-0.3333333333333333 * Float64(y / z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -0.3333333333333333 * (y / z);
end

Wolfram

code[x_, y_, z_, t_] := N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]

   2. lower-/.f64 34.9

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{z}} \]

5. Applied rewrites 34.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y}{z}} \]
6. Add Preprocessing

Developer Target 1: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024248 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y)))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))