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

Percentage Accurate: 95.5% → 97.6%

Time: 12.1s

Alternatives: 15

Speedup: 1.4×

• 27.6% 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: 95.5% 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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -1e+19)
   (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))
   (+ x (/ (- (/ t y) y) (* z 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -1e+19) {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	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 ((z * 3.0d0) <= (-1d+19)) then
        tmp = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
    else
        tmp = x + (((t / y) - y) / (z * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -1e+19) {
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= -1e+19:
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))
	else:
		tmp = x + (((t / y) - y) / (z * 3.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+19)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)));
	else
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= -1e+19)
		tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	else
		tmp = x + (((t / y) - y) / (z * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+19], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1e19

   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

   if -1e19 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x + \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -1e+19)
   (fma (/ y z) -0.3333333333333333 (+ x (/ t (* (* z 3.0) y))))
   (+ x (/ (- (/ t y) y) (* z 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -1e+19) {
		tmp = fma((y / z), -0.3333333333333333, (x + (t / ((z * 3.0) * y))));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+19)
		tmp = fma(Float64(y / z), -0.3333333333333333, Float64(x + Float64(t / Float64(Float64(z * 3.0) * y))));
	else
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+19], N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + N[(x + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1e19

   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

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-+l+ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. div-inv N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 99.7

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   if -1e19 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x + \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -1e+19)
   (fma (/ t (* z y)) 0.3333333333333333 (fma y (/ -0.3333333333333333 z) x))
   (+ x (/ (- (/ t y) y) (* z 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -1e+19) {
		tmp = fma((t / (z * y)), 0.3333333333333333, fma(y, (-0.3333333333333333 / z), x));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+19)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, fma(y, Float64(-0.3333333333333333 / z), x));
	else
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+19], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1e19

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 99.7

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

      19. lift--.f64 N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   4. Applied egg-rr 99.7%

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

   if -1e19 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 91.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -1600:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -1600.0)
     t_1
     (if (<= y 3.2e+31) (fma (/ t z) (/ 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -1600.0) {
		tmp = t_1;
	} else if (y <= 3.2e+31) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -1600.0)
		tmp = t_1;
	elseif (y <= 3.2e+31)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = t_1;
	end
	return 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[y, -1600.0], t$95$1, If[LessEqual[y, 3.2e+31], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1600 or 3.2000000000000001e31 < y

   1. Initial program 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. mul-1-neg N/A

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

      15. *-inverses N/A

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

      16. cancel-sign-sub N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 96.0

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

   7. Simplified 96.0%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. neg-mul-1 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 96.1

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

   9. Applied egg-rr 96.1%

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

   if -1600 < y < 3.2000000000000001e31

   1. Initial program 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 93.7

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

   4. Applied egg-rr 93.7%

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

   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. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      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. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. associate-/l* N/A

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

   7. Simplified 87.9%

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

      1. lift-*.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 94.6

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1600:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.65e+25)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 3.6e+33)
     (fma t (/ 0.3333333333333333 (* z y)) x)
     (+ x (/ y (* z -3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.65e+25) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 3.6e+33) {
		tmp = fma(t, (0.3333333333333333 / (z * y)), x);
	} else {
		tmp = x + (y / (z * -3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.65e+25)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 3.6e+33)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(z * y)), x);
	else
		tmp = Float64(x + Float64(y / Float64(z * -3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.65e+25], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.6e+33], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6500000000000001e25

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. mul-1-neg N/A

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

      15. *-inverses N/A

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

      16. cancel-sign-sub N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 98.6

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

   7. Simplified 98.6%

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

   if -1.6500000000000001e25 < y < 3.6000000000000003e33

   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

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   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. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      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. +-commutative N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-neg-in N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. associate-/l* N/A

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

   7. Simplified 87.0%

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

   if 3.6000000000000003e33 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. mul-1-neg N/A

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

      15. *-inverses N/A

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

      16. cancel-sign-sub N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 96.3

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

   7. Simplified 96.3%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. neg-mul-1 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 96.5

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

   9. Applied egg-rr 96.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -3.1e-12)
     t_1
     (if (<= y 9.8e-67) (/ (* t 0.3333333333333333) (* z y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -3.1e-12) {
		tmp = t_1;
	} else if (y <= 9.8e-67) {
		tmp = (t * 0.3333333333333333) / (z * y);
	} else {
		tmp = t_1;
	}
	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 (y <= (-3.1d-12)) then
        tmp = t_1
    else if (y <= 9.8d-67) then
        tmp = (t * 0.3333333333333333d0) / (z * y)
    else
        tmp = t_1
    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 (y <= -3.1e-12) {
		tmp = t_1;
	} else if (y <= 9.8e-67) {
		tmp = (t * 0.3333333333333333) / (z * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / (z * -3.0))
	tmp = 0
	if y <= -3.1e-12:
		tmp = t_1
	elif y <= 9.8e-67:
		tmp = (t * 0.3333333333333333) / (z * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -3.1e-12)
		tmp = t_1;
	elseif (y <= 9.8e-67)
		tmp = Float64(Float64(t * 0.3333333333333333) / Float64(z * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / (z * -3.0));
	tmp = 0.0;
	if (y <= -3.1e-12)
		tmp = t_1;
	elseif (y <= 9.8e-67)
		tmp = (t * 0.3333333333333333) / (z * y);
	else
		tmp = t_1;
	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[y, -3.1e-12], t$95$1, If[LessEqual[y, 9.8e-67], N[(N[(t * 0.3333333333333333), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1000000000000001e-12 or 9.79999999999999987e-67 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. mul-1-neg N/A

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

      15. *-inverses N/A

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

      16. cancel-sign-sub N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 93.9

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

   7. Simplified 93.9%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. neg-mul-1 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 94.0

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

   9. Applied egg-rr 94.0%

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

   if -3.1000000000000001e-12 < y < 9.79999999999999987e-67

   1. Initial program 90.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 67.6

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

   5. Simplified 67.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-66}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -3.1e-12)
     t_1
     (if (<= y 6.6e-66) (* t (/ 0.3333333333333333 (* z y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -3.1e-12) {
		tmp = t_1;
	} else if (y <= 6.6e-66) {
		tmp = t * (0.3333333333333333 / (z * y));
	} else {
		tmp = t_1;
	}
	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 (y <= (-3.1d-12)) then
        tmp = t_1
    else if (y <= 6.6d-66) then
        tmp = t * (0.3333333333333333d0 / (z * y))
    else
        tmp = t_1
    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 (y <= -3.1e-12) {
		tmp = t_1;
	} else if (y <= 6.6e-66) {
		tmp = t * (0.3333333333333333 / (z * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / (z * -3.0))
	tmp = 0
	if y <= -3.1e-12:
		tmp = t_1
	elif y <= 6.6e-66:
		tmp = t * (0.3333333333333333 / (z * y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -3.1e-12)
		tmp = t_1;
	elseif (y <= 6.6e-66)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(z * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / (z * -3.0));
	tmp = 0.0;
	if (y <= -3.1e-12)
		tmp = t_1;
	elseif (y <= 6.6e-66)
		tmp = t * (0.3333333333333333 / (z * y));
	else
		tmp = t_1;
	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[y, -3.1e-12], t$95$1, If[LessEqual[y, 6.6e-66], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1000000000000001e-12 or 6.5999999999999998e-66 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. mul-1-neg N/A

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

      15. *-inverses N/A

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

      16. cancel-sign-sub N/A

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

      17. *-rgt-identity N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 93.9

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

   7. Simplified 93.9%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-frac-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. neg-mul-1 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 94.0

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

   9. Applied egg-rr 94.0%

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

   if -3.1000000000000001e-12 < y < 6.5999999999999998e-66

   1. Initial program 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 92.9

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.6

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

   7. Simplified 67.6%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 67.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 67.5

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

   9. Applied egg-rr 67.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-66}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   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. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-+l- N/A

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

   7. lower--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. sub-div N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 96.6

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

4. Applied egg-rr 96.6%

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

5. Final simplification 96.6%

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

Alternative 9: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-/.f64 96.4

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

5. Simplified 96.4%

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

Alternative 10: 42.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* z -3.0))))
   (if (<= y -1.26e-59) t_1 (if (<= y 2e+80) (/ (* z x) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (z * -3.0);
	double tmp;
	if (y <= -1.26e-59) {
		tmp = t_1;
	} else if (y <= 2e+80) {
		tmp = (z * x) / z;
	} else {
		tmp = t_1;
	}
	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 = y / (z * (-3.0d0))
    if (y <= (-1.26d-59)) then
        tmp = t_1
    else if (y <= 2d+80) then
        tmp = (z * x) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / (z * -3.0);
	double tmp;
	if (y <= -1.26e-59) {
		tmp = t_1;
	} else if (y <= 2e+80) {
		tmp = (z * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (z * -3.0)
	tmp = 0
	if y <= -1.26e-59:
		tmp = t_1
	elif y <= 2e+80:
		tmp = (z * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(z * -3.0))
	tmp = 0.0
	if (y <= -1.26e-59)
		tmp = t_1;
	elseif (y <= 2e+80)
		tmp = Float64(Float64(z * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (z * -3.0);
	tmp = 0.0;
	if (y <= -1.26e-59)
		tmp = t_1;
	elseif (y <= 2e+80)
		tmp = (z * x) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.26e-59], t$95$1, If[LessEqual[y, 2e+80], N[(N[(z * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2599999999999999e-59 or 2e80 < 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 y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 59.5

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

   5. Simplified 59.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 59.6

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

   7. Applied egg-rr 59.6%

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

   if -1.2599999999999999e-59 < y < 2e80

   1. Initial program 91.4%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.6

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

   8. Simplified 82.6%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 21.7

          \[\leadsto \frac{\color{blue}{x \cdot z}}{z} \]

   11. Simplified 21.7%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   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. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-+l- N/A

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

   7. lower--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. sub-div N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 96.6

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

4. Applied egg-rr 96.6%

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

5. Taylor expanded in y around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

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

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

   5. metadata-eval N/A

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

   6. associate-*l* N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. cancel-sign-sub N/A

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

   10. mul-1-neg N/A

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

   11. associate-*r/ N/A

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

   12. associate-*l/ N/A

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

   13. associate-/l* N/A

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

   14. mul-1-neg N/A

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

   15. *-inverses N/A

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

   16. cancel-sign-sub N/A

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

   17. *-rgt-identity N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-/.f64 61.5

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

7. Simplified 61.5%

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

   1. lift-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

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

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

   7. metadata-eval N/A

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

   8. times-frac N/A

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

   9. neg-mul-1 N/A

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

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

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. frac-2neg N/A

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

   14. distribute-neg-frac2 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 61.5

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

9. Applied egg-rr 61.5%

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

10. Final simplification 61.5%

    \[\leadsto x + \frac{y}{z \cdot -3} \]
11. Add Preprocessing

Alternative 12: 63.8% 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 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   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. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-+l- N/A

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

   7. lower--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/r* N/A

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

   13. sub-div N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-/.f64 96.6

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

4. Applied egg-rr 96.6%

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

5. Taylor expanded in y around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

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

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

   5. metadata-eval N/A

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

   6. associate-*l* N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

   9. cancel-sign-sub N/A

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

   10. mul-1-neg N/A

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

   11. associate-*r/ N/A

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

   12. associate-*l/ N/A

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

   13. associate-/l* N/A

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

   14. mul-1-neg N/A

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

   15. *-inverses N/A

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

   16. cancel-sign-sub N/A

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

   17. *-rgt-identity N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-/.f64 61.5

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

7. Simplified 61.5%

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

Alternative 13: 35.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 32.6

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

5. Simplified 32.6%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 32.7

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

7. Applied egg-rr 32.7%

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

Alternative 14: 34.9% 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 94.1%

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

3. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 32.6

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

5. Simplified 32.6%

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lift-/.f64 N/A

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

   4. lower-*.f64 32.6

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

7. Applied egg-rr 32.6%

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

Alternative 15: 35.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return y * (-0.3333333333333333 / 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 * ((-0.3333333333333333d0) / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

3. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 32.6

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

5. Simplified 32.6%

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

Developer Target 1: 96.3% 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 2024212 
(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))))