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

Percentage Accurate: 95.7% → 97.8%

Time: 9.9s

Alternatives: 15

Speedup: 1.0×

• 27.7% 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: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.9e-36)
   (+ (/ t (* (* 3.0 z) y)) (- x (/ y (* 3.0 z))))
   (- x (/ (/ (- y (/ t y)) z) 3.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.9e-36) {
		tmp = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	} else {
		tmp = x - (((y - (t / 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 <= (-3.9d-36)) then
        tmp = (t / ((3.0d0 * z) * y)) + (x - (y / (3.0d0 * z)))
    else
        tmp = x - (((y - (t / 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 <= -3.9e-36) {
		tmp = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	} else {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.9e-36:
		tmp = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))
	else:
		tmp = x - (((y - (t / y)) / z) / 3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.9e-36)
		tmp = Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + Float64(x - Float64(y / Float64(3.0 * z))));
	else
		tmp = Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.9e-36)
		tmp = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	else
		tmp = x - (((y - (t / y)) / z) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.9e-36], N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9000000000000001e-36

   1. Initial program 99.8%

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

   if -3.9000000000000001e-36 < t

   1. Initial program 94.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 95.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

   6. Applied rewrites 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z \cdot y}, t, \mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.6e+110)
   (fma (/ 0.3333333333333333 (* z y)) t (fma (/ -0.3333333333333333 z) y x))
   (- x (/ (/ (- y (/ t y)) z) 3.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+110) {
		tmp = fma((0.3333333333333333 / (z * y)), t, fma((-0.3333333333333333 / z), y, x));
	} else {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.6e+110)
		tmp = fma(Float64(0.3333333333333333 / Float64(z * y)), t, fma(Float64(-0.3333333333333333 / z), y, x));
	else
		tmp = Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.6e+110], N[(N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] * t + N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6e110

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 76.4%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. associate-+l- N/A

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

      11. +-commutative N/A

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

      12. div-inv N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if -2.6e110 < t

   1. Initial program 95.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 95.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.8

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

   4. Applied rewrites 95.8%

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

   6. Applied rewrites 98.6%

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

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e+164)
   (fma (/ t (* z y)) 0.3333333333333333 x)
   (- x (/ (/ (- y (/ t y)) z) 3.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+164) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e+164)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e+164], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.9999999999999995e164

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

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

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

      15. remove-double-neg N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   9. Applied rewrites 93.1%

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

   if -8.9999999999999995e164 < t

   1. Initial program 95.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 95.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   6. Applied rewrites 98.2%

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

Alternative 4: 88.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, \frac{y}{3}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8e+100)
   (fma (/ -0.3333333333333333 z) y x)
   (if (<= y 3.3e+47)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (fma (/ -1.0 z) (/ y 3.0) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8e+100) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else if (y <= 3.3e+47) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = fma((-1.0 / z), (y / 3.0), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8e+100)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	elseif (y <= 3.3e+47)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = fma(Float64(-1.0 / z), Float64(y / 3.0), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8e+100], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 3.3e+47], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(y / 3.0), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000013e100

   1. Initial program 99.9%

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.6%

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

   if -8.00000000000000013e100 < y < 3.2999999999999999e47

   1. Initial program 93.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 94.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.4

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

   4. Applied rewrites 94.4%

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

   6. Applied rewrites 92.9%

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

   7. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

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

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

      15. remove-double-neg N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   9. Applied rewrites 86.0%

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

   if 3.2999999999999999e47 < y

   1. Initial program 99.8%

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

   3. 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.9%

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

Alternative 5: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.7e+132)
   (fma (/ t (* z y)) 0.3333333333333333 x)
   (- x (/ (- y (/ t y)) (* 3.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.7e+132) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.7e+132)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.7e+132], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.70000000000000011e132

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

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

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

      15. remove-double-neg N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   9. Applied rewrites 93.7%

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

   if -3.70000000000000011e132 < t

   1. Initial program 95.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.2

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

   4. Applied rewrites 98.2%

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

Alternative 6: 94.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e+164)
   (fma (/ t (* z y)) 0.3333333333333333 x)
   (fma (/ (- y (/ t y)) z) -0.3333333333333333 x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e+164) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = fma(((y - (t / y)) / z), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e+164)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = fma(Float64(Float64(y - Float64(t / y)) / z), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e+164], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.9999999999999995e164

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

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

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

      15. remove-double-neg N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   9. Applied rewrites 93.1%

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

   if -8.9999999999999995e164 < t

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 98.0%

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

Alternative 7: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - \frac{t}{y}, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.7e+132)
   (fma (/ t (* z y)) 0.3333333333333333 x)
   (fma (- y (/ t y)) (/ -0.3333333333333333 z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.7e+132) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = fma((y - (t / y)), (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.7e+132)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = fma(Float64(y - Float64(t / y)), Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.7e+132], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.70000000000000011e132

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

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

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

      15. remove-double-neg N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   9. Applied rewrites 93.7%

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

   if -3.70000000000000011e132 < t

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

Alternative 8: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, 0.3333333333333333 \cdot y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8e+100)
   (fma (/ -0.3333333333333333 z) y x)
   (if (<= y 3.3e+47)
     (fma (/ t (* z y)) 0.3333333333333333 x)
     (fma (/ -1.0 z) (* 0.3333333333333333 y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8e+100) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else if (y <= 3.3e+47) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = fma((-1.0 / z), (0.3333333333333333 * y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8e+100)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	elseif (y <= 3.3e+47)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = fma(Float64(-1.0 / z), Float64(0.3333333333333333 * y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8e+100], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 3.3e+47], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(0.3333333333333333 * y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000013e100

   1. Initial program 99.9%

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.6%

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

   if -8.00000000000000013e100 < y < 3.2999999999999999e47

   1. Initial program 93.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. neg-mul-1 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 94.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.4

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

   4. Applied rewrites 94.4%

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

   6. Applied rewrites 92.9%

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

   7. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

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

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

      15. remove-double-neg N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   9. Applied rewrites 86.0%

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

   if 3.2999999999999999e47 < y

   1. Initial program 99.8%

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

   3. 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-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{z}, 0.3333333333333333 \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* (* 3.0 z) y))))
   (if (<= t -6.8e+92)
     t_1
     (if (<= t 3.1e+187) (fma (/ -1.0 z) (* 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if (t <= -6.8e+92) {
		tmp = t_1;
	} else if (t <= 3.1e+187) {
		tmp = fma((-1.0 / z), (0.3333333333333333 * y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t / Float64(Float64(3.0 * z) * y))
	tmp = 0.0
	if (t <= -6.8e+92)
		tmp = t_1;
	elseif (t <= 3.1e+187)
		tmp = fma(Float64(-1.0 / z), Float64(0.3333333333333333 * y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+92], t$95$1, If[LessEqual[t, 3.1e+187], N[(N[(-1.0 / z), $MachinePrecision] * N[(0.3333333333333333 * y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.7999999999999996e92 or 3.10000000000000012e187 < t

   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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.8%

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

   if -6.7999999999999996e92 < t < 3.10000000000000012e187

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.8%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* (* 3.0 z) y))))
   (if (<= t -6.8e+92)
     t_1
     (if (<= t 3.1e+187) (fma (/ -0.3333333333333333 z) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if (t <= -6.8e+92) {
		tmp = t_1;
	} else if (t <= 3.1e+187) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t / Float64(Float64(3.0 * z) * y))
	tmp = 0.0
	if (t <= -6.8e+92)
		tmp = t_1;
	elseif (t <= 3.1e+187)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+92], t$95$1, If[LessEqual[t, 3.1e+187], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.7999999999999996e92 or 3.10000000000000012e187 < t

   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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.8%

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

   if -6.7999999999999996e92 < t < 3.10000000000000012e187

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.8%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 76.8%

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

Alternative 11: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.3333333333333333}{z \cdot y} \cdot t\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ 0.3333333333333333 (* z y)) t)))
   (if (<= t -6.8e+92)
     t_1
     (if (<= t 3.1e+187) (fma (/ -0.3333333333333333 z) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.3333333333333333 / (z * y)) * t;
	double tmp;
	if (t <= -6.8e+92) {
		tmp = t_1;
	} else if (t <= 3.1e+187) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.3333333333333333 / Float64(z * y)) * t)
	tmp = 0.0
	if (t <= -6.8e+92)
		tmp = t_1;
	elseif (t <= 3.1e+187)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -6.8e+92], t$95$1, If[LessEqual[t, 3.1e+187], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.7999999999999996e92 or 3.10000000000000012e187 < t

   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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.7%

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

   if -6.7999999999999996e92 < t < 3.10000000000000012e187

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.8%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 76.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.3333333333333333}{z \cdot y} \cdot t\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{z \cdot y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 62.9

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

5. Applied rewrites 62.9%

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

7. Applied rewrites 62.9%

   \[\leadsto \mathsf{fma}\left(\frac{-1}{z}, \color{blue}{0.3333333333333333 \cdot y}, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 62.9%

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

Alternative 13: 64.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 62.9

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

5. Applied rewrites 62.9%

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

Alternative 14: 35.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 35.9%

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

10. Applied rewrites 36.0%

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

Alternative 15: 35.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 35.9%

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

10. Applied rewrites 35.9%

    \[\leadsto \frac{-0.3333333333333333}{z} \cdot y \]
11. Add Preprocessing

Developer Target 1: 96.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024294 
(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))))