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

Percentage Accurate: 95.9% → 97.7%

Time: 10.0s

Alternatives: 12

Speedup: 0.9×

• 28.0% 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: 95.9% 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: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(z \cdot y\right) \cdot 3} + x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* 3.0 z) -5000.0)
   (fma (/ -0.3333333333333333 z) y (+ (/ t (* (* z y) 3.0)) x))
   (- x (/ (- y (/ t y)) (* 3.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((3.0 * z) <= -5000.0) {
		tmp = fma((-0.3333333333333333 / z), y, ((t / ((z * y) * 3.0)) + x));
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(3.0 * z) <= -5000.0)
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(Float64(t / Float64(Float64(z * y) * 3.0)) + x));
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(3.0 * z), $MachinePrecision], -5000.0], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(N[(t / N[(N[(z * y), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -5e3

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 99.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x + \frac{t}{\left(3 \cdot z\right) \cdot y}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5e3 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 92.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(z \cdot y\right) \cdot 3} + x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -1 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* 3.0 z) -1e+64)
   (fma (/ -0.3333333333333333 z) y (+ (/ t (* (* 3.0 z) y)) x))
   (- x (/ (- y (/ t y)) (* 3.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((3.0 * z) <= -1e+64) {
		tmp = fma((-0.3333333333333333 / z), y, ((t / ((3.0 * z) * y)) + x));
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(3.0 * z) <= -1e+64)
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + x));
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(3.0 * z), $MachinePrecision], -1e+64], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1.00000000000000002e64

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 99.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -1.00000000000000002e64 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 93.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 98.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -1 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-/.f64 N/A

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

   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 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-/.f64 N/A

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

   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 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. metadata-eval 98.6

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

   14. lift--.f64 N/A

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

   15. sub-neg N/A

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

   16. lift-/.f64 N/A

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

   17. distribute-neg-frac N/A

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

   18. neg-mul-1 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. times-frac N/A

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

   22. metadata-eval N/A

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

   23. associate-*r/ N/A

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

   24. associate-*l/ N/A

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

   25. lift-/.f64 N/A

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

   26. +-commutative N/A

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

6. Applied rewrites 98.6%

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

Alternative 5: 89.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 44000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -1.25e+41)
     t_1
     (if (<= y 44000000000000.0)
       (fma (/ t (* z y)) 0.3333333333333333 x)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -1.25e+41) {
		tmp = t_1;
	} else if (y <= 44000000000000.0) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -1.25e+41)
		tmp = t_1;
	elseif (y <= 44000000000000.0)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.25e+41], t$95$1, If[LessEqual[y, 44000000000000.0], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000006e41 or 4.4e13 < y

   1. Initial program 97.2%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if -1.25000000000000006e41 < y < 4.4e13

   1. Initial program 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 95.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. remove-double-neg N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. *-rgt-identity N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

   7. Applied rewrites 87.4%

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

Alternative 6: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{-146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-46}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -2.45e-146) t_1 (if (<= y 1.72e-46) (/ t (* (* 3.0 z) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -2.45e-146) {
		tmp = t_1;
	} else if (y <= 1.72e-46) {
		tmp = t / ((3.0 * z) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -2.45e-146)
		tmp = t_1;
	elseif (y <= 1.72e-46)
		tmp = Float64(t / Float64(Float64(3.0 * z) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.45e-146], t$95$1, If[LessEqual[y, 1.72e-46], N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4500000000000002e-146 or 1.7199999999999999e-46 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -2.4500000000000002e-146 < y < 1.7199999999999999e-46

   1. Initial program 89.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

   9. Applied rewrites 63.1%

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

Alternative 7: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{-146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-46}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -2.45e-146)
     t_1
     (if (<= y 1.72e-46) (* (/ t (* z y)) 0.3333333333333333) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -2.45e-146) {
		tmp = t_1;
	} else if (y <= 1.72e-46) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -2.45e-146)
		tmp = t_1;
	elseif (y <= 1.72e-46)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.45e-146], t$95$1, If[LessEqual[y, 1.72e-46], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4500000000000002e-146 or 1.7199999999999999e-46 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -2.4500000000000002e-146 < y < 1.7199999999999999e-46

   1. Initial program 89.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 8: 95.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 97.2

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

   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 97.2

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

4. Applied rewrites 97.2%

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

Alternative 9: 95.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in t around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

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

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

   4. associate-/r* N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

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

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

   10. associate-*r/ N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.1%

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

Alternative 10: 63.3% 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 94.5%

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

3. Taylor expanded in t around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 67.8

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

5. Applied rewrites 67.8%

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

Alternative 11: 35.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 37.8

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

5. Applied rewrites 37.8%

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

7. Applied rewrites 37.8%

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

Alternative 12: 35.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 37.8

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

5. Applied rewrites 37.8%

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

6. Final simplification 37.8%

   \[\leadsto \frac{y}{z} \cdot -0.3333333333333333 \]
7. 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 2024249 
(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))))