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

Percentage Accurate: 95.9% → 97.5%

Time: 7.8s

Alternatives: 12

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: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2e+74)
   (+ (/ 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 <= -2e+74) {
		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 <= (-2d+74)) 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 <= -2e+74) {
		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 <= -2e+74:
		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 <= -2e+74)
		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 <= -2e+74)
		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, -2e+74], 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 < -1.9999999999999999e74

   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

   if -1.9999999999999999e74 < t

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      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 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 98.9

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

   6. Applied rewrites 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+74}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-224}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (- y (/ t y)) (* 3.0 z)))))
   (if (<= y -6.5e-72)
     t_1
     (if (<= y 4.3e-224) (fma (/ (/ t z) y) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y - (t / y)) / (3.0 * z));
	double tmp;
	if (y <= -6.5e-72) {
		tmp = t_1;
	} else if (y <= 4.3e-224) {
		tmp = fma(((t / z) / y), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -6.5e-72)
		tmp = t_1;
	elseif (y <= 4.3e-224)
		tmp = fma(Float64(Float64(t / z) / y), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-72], t$95$1, If[LessEqual[y, 4.3e-224], N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999997e-72 or 4.3e-224 < y

   1. Initial program 96.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. 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 98.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if -6.4999999999999997e-72 < y < 4.3e-224

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. neg-mul-1 N/A

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

      15. *-inverses N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 98.4%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{0.3333333333333333 \cdot \left(y - \frac{t}{y}\right)}{z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (* 0.3333333333333333 (- y (/ t y))) z))))
   (if (<= y -6.5e-72)
     t_1
     (if (<= y 2e-95) (fma (/ (/ t z) y) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((0.3333333333333333 * (y - (t / y))) / z);
	double tmp;
	if (y <= -6.5e-72) {
		tmp = t_1;
	} else if (y <= 2e-95) {
		tmp = fma(((t / z) / y), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(0.3333333333333333 * Float64(y - Float64(t / y))) / z))
	tmp = 0.0
	if (y <= -6.5e-72)
		tmp = t_1;
	elseif (y <= 2e-95)
		tmp = fma(Float64(Float64(t / z) / y), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(0.3333333333333333 * N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-72], t$95$1, If[LessEqual[y, 2e-95], N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4999999999999997e-72 or 1.99999999999999998e-95 < y

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -6.4999999999999997e-72 < y < 1.99999999999999998e-95

   1. Initial program 91.2%

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

   3. Taylor expanded in y around 0

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

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

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. neg-mul-1 N/A

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

      15. *-inverses N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 96.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot \left(y - \frac{t}{y}\right)}{z}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot \left(y - \frac{t}{y}\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{3 \cdot z}\\ \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* 3.0 z)))))
   (if (<= y -1400000000000.0)
     t_1
     (if (<= y 1.5e+17) (fma (/ (/ t z) y) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = t_1;
	} else if (y <= 1.5e+17) {
		tmp = fma(((t / z) / y), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -1400000000000.0)
		tmp = t_1;
	elseif (y <= 1.5e+17)
		tmp = fma(Float64(Float64(t / z) / y), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1400000000000.0], t$95$1, If[LessEqual[y, 1.5e+17], N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e12 or 1.5e17 < y

   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. cancel-sub-sign-inv N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. div-add N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.7%

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

   if -1.4e12 < y < 1.5e17

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. neg-mul-1 N/A

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

      15. *-inverses N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.5%

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

Alternative 5: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{3 \cdot z}\\ \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* 3.0 z)))))
   (if (<= y -1400000000000.0)
     t_1
     (if (<= y 1.5e+17) (fma (/ t z) (/ 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = t_1;
	} else if (y <= 1.5e+17) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -1400000000000.0)
		tmp = t_1;
	elseif (y <= 1.5e+17)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1400000000000.0], t$95$1, If[LessEqual[y, 1.5e+17], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e12 or 1.5e17 < y

   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. cancel-sub-sign-inv N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. div-add N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.7%

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

   if -1.4e12 < y < 1.5e17

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. neg-mul-1 N/A

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

      15. *-inverses N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 94.4%

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

Alternative 6: 90.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{3 \cdot z}\\ \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* 3.0 z)))))
   (if (<= y -1400000000000.0)
     t_1
     (if (<= y 1.5e+17) (fma (/ t (* z y)) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = t_1;
	} else if (y <= 1.5e+17) {
		tmp = fma((t / (z * y)), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -1400000000000.0)
		tmp = t_1;
	elseif (y <= 1.5e+17)
		tmp = fma(Float64(t / Float64(z * y)), 0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1400000000000.0], t$95$1, If[LessEqual[y, 1.5e+17], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e12 or 1.5e17 < y

   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. cancel-sub-sign-inv N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. div-add N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.7%

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

   if -1.4e12 < y < 1.5e17

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. neg-mul-1 N/A

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

      15. *-inverses N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

Alternative 7: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{3 \cdot z}\\ \mathbf{if}\;y \leq -1400000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{z \cdot y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* 3.0 z)))))
   (if (<= y -1400000000000.0)
     t_1
     (if (<= y 1.5e+17) (fma t (/ 0.3333333333333333 (* z y)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (y <= -1400000000000.0) {
		tmp = t_1;
	} else if (y <= 1.5e+17) {
		tmp = fma(t, (0.3333333333333333 / (z * y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -1400000000000.0)
		tmp = t_1;
	elseif (y <= 1.5e+17)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(z * y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1400000000000.0], t$95$1, If[LessEqual[y, 1.5e+17], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4e12 or 1.5e17 < y

   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. cancel-sub-sign-inv N/A

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. div-add N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.7%

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

   if -1.4e12 < y < 1.5e17

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. neg-mul-1 N/A

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

      15. *-inverses N/A

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

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

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 89.9%

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

Alternative 8: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{3 \cdot z}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-183}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* 3.0 z)))))
   (if (<= y -7e-183)
     t_1
     (if (<= y 2.7e-167) (* (/ t (* z y)) 0.3333333333333333) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (y <= -7e-183) {
		tmp = t_1;
	} else if (y <= 2.7e-167) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (3.0d0 * z))
    if (y <= (-7d-183)) then
        tmp = t_1
    else if (y <= 2.7d-167) then
        tmp = (t / (z * y)) * 0.3333333333333333d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (3.0 * z));
	double tmp;
	if (y <= -7e-183) {
		tmp = t_1;
	} else if (y <= 2.7e-167) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / (3.0 * z))
	tmp = 0
	if y <= -7e-183:
		tmp = t_1
	elif y <= 2.7e-167:
		tmp = (t / (z * y)) * 0.3333333333333333
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -7e-183)
		tmp = t_1;
	elseif (y <= 2.7e-167)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (3.0 * z));
	tmp = 0.0;
	if (y <= -7e-183)
		tmp = t_1;
	elseif (y <= 2.7e-167)
		tmp = (t / (z * y)) * 0.3333333333333333;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-183], t$95$1, If[LessEqual[y, 2.7e-167], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999983e-183 or 2.7000000000000001e-167 < y

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. distribute-lft-in N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. *-inverses N/A

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

      13. associate-/l* N/A

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

      14. div-add N/A

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

      15. distribute-rgt-in N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. *-rgt-identity N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 85.2%

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

   if -6.99999999999999983e-183 < y < 2.7000000000000001e-167

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

Alternative 9: 63.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in y around inf

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

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

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. div-add N/A

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

   15. distribute-rgt-in N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. *-inverses N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-/.f64 71.1

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

5. Applied rewrites 71.1%

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

7. Applied rewrites 71.1%

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

9. Applied rewrites 71.2%

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

Alternative 10: 63.6% 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 96.7%

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

3. Taylor expanded in y around inf

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

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

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. div-add N/A

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

   15. distribute-rgt-in N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. *-inverses N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-/.f64 71.1

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

5. Applied rewrites 71.1%

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

7. Applied rewrites 71.1%

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

9. Applied rewrites 71.1%

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

Alternative 11: 63.5% 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 96.7%

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

3. Taylor expanded in y around inf

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

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

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. div-add N/A

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

   15. distribute-rgt-in N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. *-inverses N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-/.f64 71.1

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

5. Applied rewrites 71.1%

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

Alternative 12: 35.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in y around inf

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

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

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. *-inverses N/A

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

   13. associate-/l* N/A

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

   14. div-add N/A

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

   15. distribute-rgt-in N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. *-inverses N/A

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

   19. *-rgt-identity N/A

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

   20. +-commutative N/A

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

   21. lower-fma.f64 N/A

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

   22. lower-/.f64 71.1

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

5. Applied rewrites 71.1%

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

7. Applied rewrites 71.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 35.0%

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

Developer Target 1: 96.2% 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 2024298 
(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))))