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

Percentage Accurate: 95.8% → 96.0%

Time: 5.9s

Alternatives: 14

Speedup: 1.0×

• 28.3% 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: 96.0% accurate, 0.9× speedup

Math

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

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(Float64(t / z) / Float64(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 / z), $MachinePrecision] / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. lift-/.f64 N/A

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

   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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t\_1 + \frac{t}{\left(z \cdot 3\right) \cdot y} \leq -5 \cdot 10^{+275}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{t}{\left(3 \cdot y\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (+ t_1 (/ t (* (* z 3.0) y))) -5e+275)
     (- x (/ (- y (/ t y)) (* 3.0 z)))
     (+ t_1 (/ t (* (* 3.0 y) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / ((z * 3.0) * y))) <= -5e+275) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = t_1 + (t / ((3.0 * y) * 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 / (z * 3.0d0))
    if ((t_1 + (t / ((z * 3.0d0) * y))) <= (-5d+275)) then
        tmp = x - ((y - (t / y)) / (3.0d0 * z))
    else
        tmp = t_1 + (t / ((3.0d0 * y) * 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 / (z * 3.0));
	double tmp;
	if ((t_1 + (t / ((z * 3.0) * y))) <= -5e+275) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = t_1 + (t / ((3.0 * y) * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / (z * 3.0))
	tmp = 0
	if (t_1 + (t / ((z * 3.0) * y))) <= -5e+275:
		tmp = x - ((y - (t / y)) / (3.0 * z))
	else:
		tmp = t_1 + (t / ((3.0 * y) * z))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(t / Float64(Float64(z * 3.0) * y))) <= -5e+275)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = Float64(t_1 + Float64(t / Float64(Float64(3.0 * y) * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (z * 3.0));
	tmp = 0.0;
	if ((t_1 + (t / ((z * 3.0) * y))) <= -5e+275)
		tmp = x - ((y - (t / y)) / (3.0 * z));
	else
		tmp = t_1 + (t / ((3.0 * y) * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+275], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t / N[(N[(3.0 * y), $MachinePrecision] * 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))) < -5.0000000000000003e275

   1. Initial program 85.5%

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

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

   if -5.0000000000000003e275 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 97.5%

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

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-126} \lor \neg \left(y \leq 2.05 \cdot 10^{-148}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.7e-126) (not (<= y 2.05e-148)))
   (- x (/ (- y (/ t y)) (* 3.0 z)))
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.7e-126) || !(y <= 2.05e-148)) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.7e-126) || !(y <= 2.05e-148))
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.7e-126], N[Not[LessEqual[y, 2.05e-148]], $MachinePrecision]], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6999999999999999e-126 or 2.0500000000000001e-148 < 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. 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 98.7

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

      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 98.7

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

   4. Applied rewrites 98.7%

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

   if -3.6999999999999999e-126 < y < 2.0500000000000001e-148

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-126} \lor \neg \left(y \leq 2.05 \cdot 10^{-148}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-126} \lor \neg \left(y \leq 2.05 \cdot 10^{-148}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.7e-126) (not (<= y 2.05e-148)))
   (fma (/ (- (/ t y) y) z) 0.3333333333333333 x)
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.7e-126) || !(y <= 2.05e-148)) {
		tmp = fma((((t / y) - y) / z), 0.3333333333333333, x);
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.7e-126) || !(y <= 2.05e-148))
		tmp = fma(Float64(Float64(Float64(t / y) - y) / z), 0.3333333333333333, x);
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.7e-126], N[Not[LessEqual[y, 2.05e-148]], $MachinePrecision]], N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6999999999999999e-126 or 2.0500000000000001e-148 < 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 x 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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -3.6999999999999999e-126 < y < 2.0500000000000001e-148

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-126} \lor \neg \left(y \leq 2.05 \cdot 10^{-148}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{y} - y}{z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e-113) (not (<= y 1.45e+26)))
   (- x (/ (/ y z) 3.0))
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e-113) || !(y <= 1.45e+26)) {
		tmp = x - ((y / z) / 3.0);
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e-113) || !(y <= 1.45e+26))
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e-113], N[Not[LessEqual[y, 1.45e+26]], $MachinePrecision]], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000008e-113 or 1.45e26 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate--r- N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift--.f64 99.1

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 92.8

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

   9. Applied rewrites 92.8%

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

   if -5.80000000000000008e-113 < y < 1.45e26

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{z}}{y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e-113) (not (<= y 1.45e+26)))
   (- x (/ (/ y z) 3.0))
   (fma 0.3333333333333333 (/ (/ t z) y) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e-113) || !(y <= 1.45e+26)) {
		tmp = x - ((y / z) / 3.0);
	} else {
		tmp = fma(0.3333333333333333, ((t / z) / y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e-113) || !(y <= 1.45e+26))
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	else
		tmp = fma(0.3333333333333333, Float64(Float64(t / z) / y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e-113], N[Not[LessEqual[y, 1.45e+26]], $MachinePrecision]], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000008e-113 or 1.45e26 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate--r- N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift--.f64 99.1

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 92.8

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

   9. Applied rewrites 92.8%

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

   if -5.80000000000000008e-113 < y < 1.45e26

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 94.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{\frac{t}{z}}{y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e-113) (not (<= y 1.45e+26)))
   (- x (/ (/ y z) 3.0))
   (fma (/ t (* z y)) 0.3333333333333333 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e-113) || !(y <= 1.45e+26)) {
		tmp = x - ((y / z) / 3.0);
	} else {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e-113) || !(y <= 1.45e+26))
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	else
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e-113], N[Not[LessEqual[y, 1.45e+26]], $MachinePrecision]], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000008e-113 or 1.45e26 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate--r- N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift--.f64 99.1

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 92.8

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

   9. Applied rewrites 92.8%

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

   if -5.80000000000000008e-113 < y < 1.45e26

   1. Initial program 91.3%

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

      1. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-inverses N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e-113) (not (<= y 1.45e+26)))
   (fma -0.3333333333333333 (/ y z) x)
   (fma (/ t (* z y)) 0.3333333333333333 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e-113) || !(y <= 1.45e+26)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e-113) || !(y <= 1.45e+26))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e-113], N[Not[LessEqual[y, 1.45e+26]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.80000000000000008e-113 or 1.45e26 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

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

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -5.80000000000000008e-113 < y < 1.45e26

   1. Initial program 91.3%

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

      1. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-inverses N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

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

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-113} \lor \neg \left(y \leq 1.45 \cdot 10^{+26}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-113} \lor \neg \left(y \leq 4.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.85e-113) (not (<= y 4.5e-143)))
   (fma -0.3333333333333333 (/ y z) x)
   (/ (* 0.3333333333333333 t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.85e-113) || !(y <= 4.5e-143)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = (0.3333333333333333 * t) / (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.85e-113) || !(y <= 4.5e-143))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = Float64(Float64(0.3333333333333333 * t) / Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.85e-113], N[Not[LessEqual[y, 4.5e-143]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.3333333333333333 * t), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.85000000000000014e-113 or 4.5e-143 < y

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

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

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -3.85000000000000014e-113 < y < 4.5e-143

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-113} \lor \neg \left(y \leq 4.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-113} \lor \neg \left(y \leq 4.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.85e-113) (not (<= y 4.5e-143)))
   (fma -0.3333333333333333 (/ y z) x)
   (* (/ t (* z y)) 0.3333333333333333)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.85e-113) || !(y <= 4.5e-143)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = (t / (z * y)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.85e-113) || !(y <= 4.5e-143))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.85e-113], N[Not[LessEqual[y, 4.5e-143]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.85000000000000014e-113 or 4.5e-143 < y

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

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

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -3.85000000000000014e-113 < y < 4.5e-143

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-113} \lor \neg \left(y \leq 4.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-113} \lor \neg \left(y \leq 4.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.85e-113) (not (<= y 4.5e-143)))
   (fma -0.3333333333333333 (/ y z) x)
   (* t (/ 0.3333333333333333 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.85e-113) || !(y <= 4.5e-143)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = t * (0.3333333333333333 / (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.85e-113) || !(y <= 4.5e-143))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.85e-113], N[Not[LessEqual[y, 4.5e-143]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.85000000000000014e-113 or 4.5e-143 < y

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. fp-cancel-sign-sub N/A

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

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

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

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

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -3.85000000000000014e-113 < y < 4.5e-143

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-113} \lor \neg \left(y \leq 4.5 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+69} \lor \neg \left(y \leq 1.35 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -6.7e+69) (not (<= y 1.35e+57)))
   (* (/ y z) -0.3333333333333333)
   (* (/ x t) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.7e+69) || !(y <= 1.35e+57)) {
		tmp = (y / z) * -0.3333333333333333;
	} else {
		tmp = (x / t) * t;
	}
	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 ((y <= (-6.7d+69)) .or. (.not. (y <= 1.35d+57))) then
        tmp = (y / z) * (-0.3333333333333333d0)
    else
        tmp = (x / t) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6.7e+69) || !(y <= 1.35e+57)) {
		tmp = (y / z) * -0.3333333333333333;
	} else {
		tmp = (x / t) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -6.7e+69) or not (y <= 1.35e+57):
		tmp = (y / z) * -0.3333333333333333
	else:
		tmp = (x / t) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -6.7e+69) || !(y <= 1.35e+57))
		tmp = Float64(Float64(y / z) * -0.3333333333333333);
	else
		tmp = Float64(Float64(x / t) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -6.7e+69) || ~((y <= 1.35e+57)))
		tmp = (y / z) * -0.3333333333333333;
	else
		tmp = (x / t) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -6.7e+69], N[Not[LessEqual[y, 1.35e+57]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7000000000000001e69 or 1.3499999999999999e57 < y

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate--r- N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift--.f64 99.8

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.1

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

   9. Applied rewrites 75.1%

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

   if -6.7000000000000001e69 < y < 1.3499999999999999e57

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{t} \cdot t \]
   10. Step-by-step derivation

   11. Applied rewrites 27.7%

       \[\leadsto \frac{x}{t} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+69} \lor \neg \left(y \leq 1.35 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% 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 94.8%

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

3. Taylor expanded in y around inf

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

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

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. fp-cancel-sign-sub N/A

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

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

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

   12. associate-/l* N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

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

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

   16. *-inverses N/A

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

   17. *-rgt-identity N/A

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

   18. *-lft-identity N/A

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

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

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

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

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

   21. *-lft-identity N/A

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

   22. lower-fma.f64 N/A

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

   23. lower-/.f64 65.1

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

5. Applied rewrites 65.1%

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

6. Final simplification 65.1%

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

Alternative 14: 22.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{t} \cdot t \end{array} \]

FPCore

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

C

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

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 / t) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

3. Taylor expanded in y around 0

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

   1. div-add N/A

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

   2. associate-/l* N/A

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

   3. associate-/r* N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

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

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. mul-1-neg N/A

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

   10. metadata-eval N/A

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

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

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

   12. *-lft-identity N/A

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

   13. associate-*r/ N/A

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

   14. times-frac N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 53.9%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{x}{t} \cdot t \]
10. Step-by-step derivation

11. Applied rewrites 23.6%

    \[\leadsto \frac{x}{t} \cdot t \]

12. Final simplification 23.6%

    \[\leadsto \frac{x}{t} \cdot t \]
13. Add Preprocessing

Developer Target 1: 96.0% 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 2024332 
(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))))