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

Percentage Accurate: 95.7% → 97.4%

Time: 12.9s

Alternatives: 20

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: 95.7% 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: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, y \cdot 0.3333333333333333, x + \frac{t}{y \cdot \left(z \cdot 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6e+49)
   (fma (/ -1.0 z) (* y 0.3333333333333333) (+ x (/ t (* y (* z 3.0)))))
   (+ x (/ (- (/ t y) y) (* z 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6e+49) {
		tmp = fma((-1.0 / z), (y * 0.3333333333333333), (x + (t / (y * (z * 3.0)))));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6e+49)
		tmp = fma(Float64(-1.0 / z), Float64(y * 0.3333333333333333), Float64(x + Float64(t / Float64(y * Float64(z * 3.0)))));
	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[t, -6e+49], N[(N[(-1.0 / z), $MachinePrecision] * N[(y * 0.3333333333333333), $MachinePrecision] + N[(x + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $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 t < -6.0000000000000005e49

   1. Initial program 98.0%

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

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

      11. neg-mul-1 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. div-inv N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 98.1

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, y \cdot 0.3333333333333333, \color{blue}{x + \frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 98.1

          \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, y \cdot 0.3333333333333333, x + \frac{t}{\color{blue}{y \cdot \left(z \cdot 3\right)}}\right) \]

   4. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y \cdot 0.3333333333333333, x + \frac{t}{y \cdot \left(z \cdot 3\right)}\right)} \]

   if -6.0000000000000005e49 < t

   1. Initial program 94.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 97.5

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, y \cdot 0.3333333333333333, x + \frac{t}{y \cdot \left(z \cdot 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.5e-32)
   (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0))))
   (+ x (/ (- (/ t y) y) (* z 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.5e-32) {
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)));
	} 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 (t <= (-2.5d-32)) then
        tmp = (x - (y / (z * 3.0d0))) + (t / (y * (z * 3.0d0)))
    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 (t <= -2.5e-32) {
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.5e-32:
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))
	else:
		tmp = x + (((t / y) - y) / (z * 3.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.5e-32)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0))));
	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 (t <= -2.5e-32)
		tmp = (x - (y / (z * 3.0))) + (t / (y * (z * 3.0)));
	else
		tmp = x + (((t / y) - y) / (z * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.5e-32], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $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 t < -2.5e-32

   1. Initial program 98.4%

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

   if -2.5e-32 < t

   1. Initial program 93.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 97.3

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

   4. Applied egg-rr 97.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.4e+49)
   (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 (t <= -5.4e+49) {
		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 (t <= -5.4e+49)
		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[t, -5.4e+49], 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 t < -5.4000000000000002e49

   1. Initial program 98.0%

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

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

      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 97.9

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

      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 97.9%

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

   if -5.4000000000000002e49 < t

   1. Initial program 94.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 97.5

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

   4. Applied egg-rr 97.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;z \cdot 3 \leq 4 \cdot 10^{-71}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -1e-71)
   (+ x (/ -0.3333333333333333 z))
   (if (<= (* z 3.0) 4e-71) (- (/ y z)) (+ x (/ 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -1e-71) {
		tmp = x + (-0.3333333333333333 / z);
	} else if ((z * 3.0) <= 4e-71) {
		tmp = -(y / z);
	} else {
		tmp = x + (1.0 / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * 3.0d0) <= (-1d-71)) then
        tmp = x + ((-0.3333333333333333d0) / z)
    else if ((z * 3.0d0) <= 4d-71) then
        tmp = -(y / z)
    else
        tmp = x + (1.0d0 / z)
    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-71) {
		tmp = x + (-0.3333333333333333 / z);
	} else if ((z * 3.0) <= 4e-71) {
		tmp = -(y / z);
	} else {
		tmp = x + (1.0 / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= -1e-71:
		tmp = x + (-0.3333333333333333 / z)
	elif (z * 3.0) <= 4e-71:
		tmp = -(y / z)
	else:
		tmp = x + (1.0 / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e-71)
		tmp = Float64(x + Float64(-0.3333333333333333 / z));
	elseif (Float64(z * 3.0) <= 4e-71)
		tmp = Float64(-Float64(y / z));
	else
		tmp = Float64(x + Float64(1.0 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= -1e-71)
		tmp = x + (-0.3333333333333333 / z);
	elseif ((z * 3.0) <= 4e-71)
		tmp = -(y / z);
	else
		tmp = x + (1.0 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e-71], N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 4e-71], (-N[(y / z), $MachinePrecision]), N[(x + N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -9.9999999999999992e-72

   1. Initial program 96.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 43.1

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 43.1%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

   if -9.9999999999999992e-72 < (*.f64 z #s(literal 3 binary64)) < 3.9999999999999997e-71

   1. Initial program 90.3%

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

   3. Taylor expanded in y around 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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 47.2

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

   5. Simplified 47.2%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.3

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

   8. Simplified 33.3%

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

   if 3.9999999999999997e-71 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 41.1

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 41.1%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{1}{z}} \]
   7. Step-by-step derivation

      1. lower-/.f64 41.4

         \[\leadsto x + \color{blue}{\frac{1}{z}} \]

   8. Simplified 41.4%

      \[\leadsto x + \color{blue}{\frac{1}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;z \cdot 3 \leq 4 \cdot 10^{-71}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot 3 \leq 4 \cdot 10^{-71}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ -0.3333333333333333 z))))
   (if (<= (* z 3.0) -1e-71) t_1 (if (<= (* z 3.0) 4e-71) (- (/ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (-0.3333333333333333 / z);
	double tmp;
	if ((z * 3.0) <= -1e-71) {
		tmp = t_1;
	} else if ((z * 3.0) <= 4e-71) {
		tmp = -(y / 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 = x + ((-0.3333333333333333d0) / z)
    if ((z * 3.0d0) <= (-1d-71)) then
        tmp = t_1
    else if ((z * 3.0d0) <= 4d-71) then
        tmp = -(y / 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 = x + (-0.3333333333333333 / z);
	double tmp;
	if ((z * 3.0) <= -1e-71) {
		tmp = t_1;
	} else if ((z * 3.0) <= 4e-71) {
		tmp = -(y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (-0.3333333333333333 / z)
	tmp = 0
	if (z * 3.0) <= -1e-71:
		tmp = t_1
	elif (z * 3.0) <= 4e-71:
		tmp = -(y / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(-0.3333333333333333 / z))
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e-71)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 4e-71)
		tmp = Float64(-Float64(y / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (-0.3333333333333333 / z);
	tmp = 0.0;
	if ((z * 3.0) <= -1e-71)
		tmp = t_1;
	elseif ((z * 3.0) <= 4e-71)
		tmp = -(y / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e-71], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 4e-71], (-N[(y / z), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -9.9999999999999992e-72 or 3.9999999999999997e-71 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 98.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 42.2

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 42.2%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

   if -9.9999999999999992e-72 < (*.f64 z #s(literal 3 binary64)) < 3.9999999999999997e-71

   1. Initial program 90.3%

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

   3. Taylor expanded in y around 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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 47.2

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

   5. Simplified 47.2%

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

   6. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 33.3

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

   8. Simplified 33.3%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;z \cdot 3 \leq 4 \cdot 10^{-71}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-247}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-8)
   (/ (* y -0.3333333333333333) z)
   (if (<= y 7.6e-247)
     (+ x (/ -0.3333333333333333 z))
     (if (<= y 7.5e+42)
       (fma t (/ -0.3333333333333333 y) x)
       (/ y (* z -3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-8) {
		tmp = (y * -0.3333333333333333) / z;
	} else if (y <= 7.6e-247) {
		tmp = x + (-0.3333333333333333 / z);
	} else if (y <= 7.5e+42) {
		tmp = fma(t, (-0.3333333333333333 / y), x);
	} else {
		tmp = y / (z * -3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-8)
		tmp = Float64(Float64(y * -0.3333333333333333) / z);
	elseif (y <= 7.6e-247)
		tmp = Float64(x + Float64(-0.3333333333333333 / z));
	elseif (y <= 7.5e+42)
		tmp = fma(t, Float64(-0.3333333333333333 / y), x);
	else
		tmp = Float64(y / Float64(z * -3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-8], N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 7.6e-247], N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+42], N[(t * N[(-0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.99999999999999986e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   if -8.99999999999999986e-8 < y < 7.59999999999999977e-247

   1. Initial program 90.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 35.1

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 35.1%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

   if 7.59999999999999977e-247 < y < 7.50000000000000041e42

   1. Initial program 95.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 93.5

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

   4. Applied egg-rr 93.5%

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

   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. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-/l/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-out N/A

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

      17. remove-double-neg N/A

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

      18. associate-*l/ N/A

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

      19. associate-/l* N/A

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

      20. *-inverses N/A

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

      21. *-rgt-identity N/A

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

   7. Simplified 91.9%

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

   8. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{\frac{-1}{3}}{y}}, x\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 45.7

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

   10. Simplified 45.7%

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

   if 7.50000000000000041e42 < y

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 82.6

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

   5. Simplified 82.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.7

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

   7. Applied egg-rr 82.7%

      \[\leadsto \color{blue}{\frac{y}{-3 \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-247}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{z \cdot 3}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.4e-7)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 2e+42)
     (fma t (/ 0.3333333333333333 (* z y)) x)
     (fma -1.0 (/ y (* z 3.0)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.4e-7) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 2e+42) {
		tmp = fma(t, (0.3333333333333333 / (z * y)), x);
	} else {
		tmp = fma(-1.0, (y / (z * 3.0)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.4e-7)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 2e+42)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(z * y)), x);
	else
		tmp = fma(-1.0, Float64(y / Float64(z * 3.0)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.4e-7], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2e+42], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(-1.0 * N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.39999999999999974e-7

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. 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-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 93.4

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

   5. Simplified 93.4%

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

   if -3.39999999999999974e-7 < y < 2.00000000000000009e42

   1. Initial program 92.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 92.2

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

   4. Applied egg-rr 92.2%

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

   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. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-/l/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. *-lft-identity N/A

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

      12. associate-*l/ N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-neg-out N/A

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

      17. remove-double-neg N/A

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

      18. associate-*l/ N/A

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

      19. associate-/l* N/A

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

      20. *-inverses N/A

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

      21. *-rgt-identity N/A

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

   7. Simplified 90.0%

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

   if 2.00000000000000009e42 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. 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-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 95.8

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

   5. Simplified 95.8%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 95.9

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

   7. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{y}{z \cdot 3}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{z \cdot 3}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{z \cdot 3}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1e-124)
   (fma y (/ -0.3333333333333333 z) x)
   (if (<= y 7.6e-13) (/ t (* 3.0 (* z y))) (fma -1.0 (/ y (* z 3.0)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e-124) {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	} else if (y <= 7.6e-13) {
		tmp = t / (3.0 * (z * y));
	} else {
		tmp = fma(-1.0, (y / (z * 3.0)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1e-124)
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 7.6e-13)
		tmp = Float64(t / Float64(3.0 * Float64(z * y)));
	else
		tmp = fma(-1.0, Float64(y / Float64(z * 3.0)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1e-124], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 7.6e-13], N[(t / N[(3.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.99999999999999933e-125

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. 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-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 80.7

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

   5. Simplified 80.7%

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

   if -9.99999999999999933e-125 < y < 7.5999999999999999e-13

   1. Initial program 90.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 91.1

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

   4. Applied egg-rr 91.1%

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

   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 63.2

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

   7. Simplified 63.2%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-/.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 64.8

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

   9. Applied egg-rr 64.8%

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

       1. lift-*.f64 N/A

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

       2. associate-/l/ N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. div-inv N/A

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

       7. associate-/l* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. div-inv N/A

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

       11. metadata-eval N/A

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

       12. lower-*.f64 63.2

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 63.2

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

   11. Applied egg-rr 63.2%

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

   if 7.5999999999999999e-13 < 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

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. 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-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 90.8

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

   5. Simplified 90.8%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*l/ N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 90.9

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

   7. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{y}{z \cdot 3}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{y}{z \cdot 3}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1e-124) t_1 (if (<= y 7.6e-13) (/ t (* 3.0 (* z y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1e-124) {
		tmp = t_1;
	} else if (y <= 7.6e-13) {
		tmp = t / (3.0 * (z * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1e-124)
		tmp = t_1;
	elseif (y <= 7.6e-13)
		tmp = Float64(t / Float64(3.0 * Float64(z * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1e-124], t$95$1, If[LessEqual[y, 7.6e-13], N[(t / N[(3.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999933e-125 or 7.5999999999999999e-13 < y

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. 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-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 85.1

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

   5. Simplified 85.1%

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

   if -9.99999999999999933e-125 < y < 7.5999999999999999e-13

   1. Initial program 90.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 91.1

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

   4. Applied egg-rr 91.1%

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

   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 63.2

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

   7. Simplified 63.2%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-/.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 64.8

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

   9. Applied egg-rr 64.8%

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

       1. lift-*.f64 N/A

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

       2. associate-/l/ N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. div-inv N/A

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

       7. associate-/l* N/A

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

       8. lift-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. div-inv N/A

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

       11. metadata-eval N/A

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

       12. lower-*.f64 63.2

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 63.2

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

   11. Applied egg-rr 63.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1e-124)
     t_1
     (if (<= y 7.6e-13) (* t (/ 0.3333333333333333 (* z y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1e-124) {
		tmp = t_1;
	} else if (y <= 7.6e-13) {
		tmp = t * (0.3333333333333333 / (z * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1e-124)
		tmp = t_1;
	elseif (y <= 7.6e-13)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(z * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1e-124], t$95$1, If[LessEqual[y, 7.6e-13], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999933e-125 or 7.5999999999999999e-13 < y

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
   4. 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-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 85.1

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

   5. Simplified 85.1%

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

   if -9.99999999999999933e-125 < y < 7.5999999999999999e-13

   1. Initial program 90.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 91.1

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

   4. Applied egg-rr 91.1%

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

   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 63.2

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

   7. Simplified 63.2%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 63.1

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

   9. Applied egg-rr 63.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-124}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.9% 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 95.1%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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 95.4

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

4. Applied egg-rr 95.4%

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

5. Final simplification 95.4%

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

Alternative 12: 95.8% 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 95.1%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\left(\frac{1}{3} \cdot \frac{t}{{y}^{2} \cdot z} + \frac{x}{y}\right) - \frac{1}{3} \cdot \frac{1}{z}\right)} \]

5. Simplified 95.4%

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

Alternative 13: 47.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-8)
   (/ (* y -0.3333333333333333) z)
   (if (<= y 7.5e+42) (+ x (/ -0.3333333333333333 z)) (/ y (* z -3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-8) {
		tmp = (y * -0.3333333333333333) / z;
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / z);
	} else {
		tmp = 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 (y <= (-9d-8)) then
        tmp = (y * (-0.3333333333333333d0)) / z
    else if (y <= 7.5d+42) then
        tmp = x + ((-0.3333333333333333d0) / z)
    else
        tmp = 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 (y <= -9e-8) {
		tmp = (y * -0.3333333333333333) / z;
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / z);
	} else {
		tmp = y / (z * -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9e-8:
		tmp = (y * -0.3333333333333333) / z
	elif y <= 7.5e+42:
		tmp = x + (-0.3333333333333333 / z)
	else:
		tmp = y / (z * -3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-8)
		tmp = Float64(Float64(y * -0.3333333333333333) / z);
	elseif (y <= 7.5e+42)
		tmp = Float64(x + Float64(-0.3333333333333333 / z));
	else
		tmp = Float64(y / Float64(z * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9e-8)
		tmp = (y * -0.3333333333333333) / z;
	elseif (y <= 7.5e+42)
		tmp = x + (-0.3333333333333333 / z);
	else
		tmp = y / (z * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-8], N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 7.5e+42], N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999986e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

   if -8.99999999999999986e-8 < y < 7.50000000000000041e42

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 35.2

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 35.2%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

   if 7.50000000000000041e42 < y

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 82.6

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

   5. Simplified 82.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.7

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

   7. Applied egg-rr 82.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1e-7)
   (* y (/ -0.3333333333333333 z))
   (if (<= y 7.5e+42) (+ x (/ -0.3333333333333333 z)) (/ y (* z -3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1e-7) {
		tmp = y * (-0.3333333333333333 / z);
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / z);
	} else {
		tmp = 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 (y <= (-1d-7)) then
        tmp = y * ((-0.3333333333333333d0) / z)
    else if (y <= 7.5d+42) then
        tmp = x + ((-0.3333333333333333d0) / z)
    else
        tmp = 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 (y <= -1e-7) {
		tmp = y * (-0.3333333333333333 / z);
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / z);
	} else {
		tmp = y / (z * -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1e-7:
		tmp = y * (-0.3333333333333333 / z)
	elif y <= 7.5e+42:
		tmp = x + (-0.3333333333333333 / z)
	else:
		tmp = y / (z * -3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	elseif (y <= 7.5e+42)
		tmp = Float64(x + Float64(-0.3333333333333333 / z));
	else
		tmp = Float64(y / Float64(z * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1e-7)
		tmp = y * (-0.3333333333333333 / z);
	elseif (y <= 7.5e+42)
		tmp = x + (-0.3333333333333333 / z);
	else
		tmp = y / (z * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1e-7], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+42], N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

      1. associate-/l* N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 76.9

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

   7. Applied egg-rr 76.9%

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

   if -9.9999999999999995e-8 < y < 7.50000000000000041e42

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 35.2

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 35.2%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

   if 7.50000000000000041e42 < y

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 82.6

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

   5. Simplified 82.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.7

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

   7. Applied egg-rr 82.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ -0.3333333333333333 z))))
   (if (<= y -1e-7)
     t_1
     (if (<= y 7.5e+42) (+ x (/ -0.3333333333333333 z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (-0.3333333333333333 / z);
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1;
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / 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 * ((-0.3333333333333333d0) / z)
    if (y <= (-1d-7)) then
        tmp = t_1
    else if (y <= 7.5d+42) then
        tmp = x + ((-0.3333333333333333d0) / 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 * (-0.3333333333333333 / z);
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1;
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (-0.3333333333333333 / z)
	tmp = 0
	if y <= -1e-7:
		tmp = t_1
	elif y <= 7.5e+42:
		tmp = x + (-0.3333333333333333 / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(-0.3333333333333333 / z))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = t_1;
	elseif (y <= 7.5e+42)
		tmp = Float64(x + Float64(-0.3333333333333333 / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (-0.3333333333333333 / z);
	tmp = 0.0;
	if (y <= -1e-7)
		tmp = t_1;
	elseif (y <= 7.5e+42)
		tmp = x + (-0.3333333333333333 / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], t$95$1, If[LessEqual[y, 7.5e+42], N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999995e-8 or 7.50000000000000041e42 < y

   1. Initial program 98.9%

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

   3. 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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 79.6

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

   5. Simplified 79.6%

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

      1. associate-/l* N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 79.6

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

   7. Applied egg-rr 79.6%

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

   if -9.9999999999999995e-8 < y < 7.50000000000000041e42

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 35.2

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 35.2%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -0.3333333333333333 (/ y z))))
   (if (<= y -1e-7)
     t_1
     (if (<= y 7.5e+42) (+ x (/ -0.3333333333333333 z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.3333333333333333 * (y / z);
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1;
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / 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 = (-0.3333333333333333d0) * (y / z)
    if (y <= (-1d-7)) then
        tmp = t_1
    else if (y <= 7.5d+42) then
        tmp = x + ((-0.3333333333333333d0) / 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 = -0.3333333333333333 * (y / z);
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1;
	} else if (y <= 7.5e+42) {
		tmp = x + (-0.3333333333333333 / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.3333333333333333 * (y / z)
	tmp = 0
	if y <= -1e-7:
		tmp = t_1
	elif y <= 7.5e+42:
		tmp = x + (-0.3333333333333333 / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-0.3333333333333333 * Float64(y / z))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = t_1;
	elseif (y <= 7.5e+42)
		tmp = Float64(x + Float64(-0.3333333333333333 / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.3333333333333333 * (y / z);
	tmp = 0.0;
	if (y <= -1e-7)
		tmp = t_1;
	elseif (y <= 7.5e+42)
		tmp = x + (-0.3333333333333333 / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], t$95$1, If[LessEqual[y, 7.5e+42], N[(x + N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999995e-8 or 7.50000000000000041e42 < y

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      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 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 79.5

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

   7. Simplified 79.5%

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

   if -9.9999999999999995e-8 < y < 7.50000000000000041e42

   1. Initial program 92.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 35.2

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

   5. Simplified 35.2%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 64.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
4. 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-lft-in N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 62.1

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

5. Simplified 62.1%

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

Alternative 18: 18.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 36.3

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

5. Simplified 36.3%

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

6. Taylor expanded in y around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 17.4

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

8. Simplified 17.4%

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

9. Final simplification 17.4%

   \[\leadsto -\frac{y}{z} \]
10. Add Preprocessing

Alternative 19: 3.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 1.0 z))

C

double code(double x, double y, double z, double t) {
	return 1.0 / z;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 / z
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 / z;
}

Python

def code(x, y, z, t):
	return 1.0 / z

Julia

function code(x, y, z, t)
	return Float64(1.0 / z)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 / z;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 / z), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in t around 0

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

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 27.7

      \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

5. Simplified 27.7%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

6. Taylor expanded in x around 0

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

   1. lower-/.f64 3.1

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

8. Simplified 3.1%

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

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \]
10. Step-by-step derivation

    1. lower-/.f64 3.2

       \[\leadsto \color{blue}{\frac{1}{z}} \]

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{1}{z}} \]
12. Add Preprocessing

Alternative 20: 3.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -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 = (-0.3333333333333333d0) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in t around 0

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

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 27.7

      \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \]

5. Simplified 27.7%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z}} \]

6. Taylor expanded in x around 0

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

   1. lower-/.f64 3.1

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

8. Simplified 3.1%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{z}} \]
9. Add Preprocessing

Developer Target 1: 95.8% 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 2024214 
(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))))