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

Percentage Accurate: 95.5% → 95.3%

Time: 10.5s

Alternatives: 14

Speedup: 1.0×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{if}\;y \leq -6 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (- (/ t y) y) (* z 3.0)))))
   (if (<= y -6e-33)
     t_1
     (if (<= y 8.5e-100) (fma t (/ 0.3333333333333333 (* y z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((t / y) - y) / (z * 3.0));
	double tmp;
	if (y <= -6e-33) {
		tmp = t_1;
	} else if (y <= 8.5e-100) {
		tmp = fma(t, (0.3333333333333333 / (y * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)))
	tmp = 0.0
	if (y <= -6e-33)
		tmp = t_1;
	elseif (y <= 8.5e-100)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(y * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e-33], t$95$1, If[LessEqual[y, 8.5e-100], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000003e-33 or 8.50000000000000017e-100 < y

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -6.0000000000000003e-33 < y < 8.50000000000000017e-100

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. 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

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{\left(0 - y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\left(-1 \cdot \frac{x}{y}\right) \cdot y}\right) \]

      15. associate-*r/ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y\right) \]

      16. associate-*l/ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}}\right) \]

      17. associate-/l* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}}\right) \]

      18. *-inverses N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \left(-1 \cdot x\right) \cdot \color{blue}{1}\right) \]

      19. *-rgt-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{-1 \cdot x}\right) \]

      20. associate-+r- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} + 0\right) - -1 \cdot x} \]

   7. Applied rewrites 97.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-33}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)))
   (if (<= y -6e-33)
     t_1
     (if (<= y 8.5e-100) (fma t (/ 0.3333333333333333 (* y z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((0.3333333333333333 / z), ((t / y) - y), x);
	double tmp;
	if (y <= -6e-33) {
		tmp = t_1;
	} else if (y <= 8.5e-100) {
		tmp = fma(t, (0.3333333333333333 / (y * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x)
	tmp = 0.0
	if (y <= -6e-33)
		tmp = t_1;
	elseif (y <= 8.5e-100)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(y * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -6e-33], t$95$1, If[LessEqual[y, 8.5e-100], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0000000000000003e-33 or 8.50000000000000017e-100 < y

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -6.0000000000000003e-33 < y < 8.50000000000000017e-100

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. 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

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{\left(0 - y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\left(-1 \cdot \frac{x}{y}\right) \cdot y}\right) \]

      15. associate-*r/ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y\right) \]

      16. associate-*l/ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}}\right) \]

      17. associate-/l* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}}\right) \]

      18. *-inverses N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \left(-1 \cdot x\right) \cdot \color{blue}{1}\right) \]

      19. *-rgt-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{-1 \cdot x}\right) \]

      20. associate-+r- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} + 0\right) - -1 \cdot x} \]

   7. Applied rewrites 97.8%

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

Alternative 3: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Final simplification 97.9%

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

Alternative 4: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x + \frac{t}{y \cdot \left(z \cdot 3\right)}\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return fma((y / z), -0.3333333333333333, (x + (t / (y * (z * 3.0)))));
}

Julia

function code(x, y, z, t)
	return fma(Float64(y / z), -0.3333333333333333, Float64(x + Float64(t / Float64(y * Float64(z * 3.0)))))
end

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-+l+ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. div-inv N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-+.f64 97.8

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 97.8

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

4. Applied rewrites 97.8%

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

Alternative 5: 95.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. times-frac N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 97.9

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

   19. lift--.f64 N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

4. Applied rewrites 97.8%

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

Alternative 6: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -2.2e-24)
     t_1
     (if (<= y 7.2e-93) (fma t (/ 0.3333333333333333 (* y z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -2.2e-24) {
		tmp = t_1;
	} else if (y <= 7.2e-93) {
		tmp = fma(t, (0.3333333333333333 / (y * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -2.2e-24)
		tmp = t_1;
	elseif (y <= 7.2e-93)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(y * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e-24], t$95$1, If[LessEqual[y, 7.2e-93], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000002e-24 or 7.2000000000000003e-93 < y

   1. Initial program 97.9%

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

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

   5. Applied rewrites 91.1%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if -2.20000000000000002e-24 < y < 7.2000000000000003e-93

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-+l- N/A

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

      7. lower--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. 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

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

      8. times-frac N/A

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

      9. associate-*r/ N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. neg-sub0 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \color{blue}{\left(0 - y \cdot \left(-1 \cdot \frac{x}{y}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\left(-1 \cdot \frac{x}{y}\right) \cdot y}\right) \]

      15. associate-*r/ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\frac{-1 \cdot x}{y}} \cdot y\right) \]

      16. associate-*l/ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\frac{\left(-1 \cdot x\right) \cdot y}{y}}\right) \]

      17. associate-/l* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{\left(-1 \cdot x\right) \cdot \frac{y}{y}}\right) \]

      18. *-inverses N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \left(-1 \cdot x\right) \cdot \color{blue}{1}\right) \]

      19. *-rgt-identity N/A

          \[\leadsto \frac{1}{3} \cdot \frac{t}{y \cdot z} + \left(0 - \color{blue}{-1 \cdot x}\right) \]

      20. associate-+r- N/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{t}{y \cdot z} + 0\right) - -1 \cdot x} \]

   7. Applied rewrites 97.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -2.2e-24)
     t_1
     (if (<= y 7.2e-93) (fma 0.3333333333333333 (/ t (* y z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -2.2e-24) {
		tmp = t_1;
	} else if (y <= 7.2e-93) {
		tmp = fma(0.3333333333333333, (t / (y * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -2.2e-24)
		tmp = t_1;
	elseif (y <= 7.2e-93)
		tmp = fma(0.3333333333333333, Float64(t / Float64(y * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e-24], t$95$1, If[LessEqual[y, 7.2e-93], N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000002e-24 or 7.2000000000000003e-93 < y

   1. Initial program 97.9%

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

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

   5. Applied rewrites 91.1%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

   if -2.20000000000000002e-24 < y < 7.2000000000000003e-93

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 97.8

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

   8. Applied rewrites 97.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-24}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -3.2e-102) t_1 (if (<= y 4.6e-99) (/ t (* y (* z 3.0))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -3.2e-102) {
		tmp = t_1;
	} else if (y <= 4.6e-99) {
		tmp = t / (y * (z * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y / (z * (-3.0d0)))
    if (y <= (-3.2d-102)) then
        tmp = t_1
    else if (y <= 4.6d-99) then
        tmp = t / (y * (z * 3.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -3.2e-102) {
		tmp = t_1;
	} else if (y <= 4.6e-99) {
		tmp = t / (y * (z * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / (z * -3.0))
	tmp = 0
	if y <= -3.2e-102:
		tmp = t_1
	elif y <= 4.6e-99:
		tmp = t / (y * (z * 3.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -3.2e-102)
		tmp = t_1;
	elseif (y <= 4.6e-99)
		tmp = Float64(t / Float64(y * Float64(z * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / (z * -3.0));
	tmp = 0.0;
	if (y <= -3.2e-102)
		tmp = t_1;
	elseif (y <= 4.6e-99)
		tmp = t / (y * (z * 3.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e-102], t$95$1, If[LessEqual[y, 4.6e-99], N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999986e-102 or 4.5999999999999997e-99 < y

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-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

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

   5. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if -3.19999999999999986e-102 < y < 4.5999999999999997e-99

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. div-inv N/A

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

      12. lift-/.f64 74.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y (* z -3.0)))))
   (if (<= y -3.2e-102)
     t_1
     (if (<= y 4.6e-99) (* t (/ 0.3333333333333333 (* y z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -3.2e-102) {
		tmp = t_1;
	} else if (y <= 4.6e-99) {
		tmp = t * (0.3333333333333333 / (y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y / (z * (-3.0d0)))
    if (y <= (-3.2d-102)) then
        tmp = t_1
    else if (y <= 4.6d-99) then
        tmp = t * (0.3333333333333333d0 / (y * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (y / (z * -3.0));
	double tmp;
	if (y <= -3.2e-102) {
		tmp = t_1;
	} else if (y <= 4.6e-99) {
		tmp = t * (0.3333333333333333 / (y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / (z * -3.0))
	tmp = 0
	if y <= -3.2e-102:
		tmp = t_1
	elif y <= 4.6e-99:
		tmp = t * (0.3333333333333333 / (y * z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / Float64(z * -3.0)))
	tmp = 0.0
	if (y <= -3.2e-102)
		tmp = t_1;
	elseif (y <= 4.6e-99)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(y * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / (z * -3.0));
	tmp = 0.0;
	if (y <= -3.2e-102)
		tmp = t_1;
	elseif (y <= 4.6e-99)
		tmp = t * (0.3333333333333333 / (y * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e-102], t$95$1, If[LessEqual[y, 4.6e-99], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999986e-102 or 4.5999999999999997e-99 < y

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-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

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

   5. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if -3.19999999999999986e-102 < y < 4.5999999999999997e-99

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.4

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

   7. Applied rewrites 74.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -290000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0165:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* z -3.0))))
   (if (<= y -290000000.0) t_1 (if (<= y 0.0165) (/ (* x y) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (z * -3.0);
	double tmp;
	if (y <= -290000000.0) {
		tmp = t_1;
	} else if (y <= 0.0165) {
		tmp = (x * y) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y / (z * (-3.0d0))
    if (y <= (-290000000.0d0)) then
        tmp = t_1
    else if (y <= 0.0165d0) then
        tmp = (x * y) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / (z * -3.0);
	double tmp;
	if (y <= -290000000.0) {
		tmp = t_1;
	} else if (y <= 0.0165) {
		tmp = (x * y) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (z * -3.0)
	tmp = 0
	if y <= -290000000.0:
		tmp = t_1
	elif y <= 0.0165:
		tmp = (x * y) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(z * -3.0))
	tmp = 0.0
	if (y <= -290000000.0)
		tmp = t_1;
	elseif (y <= 0.0165)
		tmp = Float64(Float64(x * y) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (z * -3.0);
	tmp = 0.0;
	if (y <= -290000000.0)
		tmp = t_1;
	elseif (y <= 0.0165)
		tmp = (x * y) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -290000000.0], t$95$1, If[LessEqual[y, 0.0165], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e8 or 0.016500000000000001 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 71.4

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

   7. Applied rewrites 71.4%

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

   if -2.9e8 < y < 0.016500000000000001

   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 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 32.0

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

   8. Applied rewrites 32.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -290000000:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 0.0165:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

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

5. Applied rewrites 66.2%

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

   1. lift-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. metadata-eval N/A

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

   15. lower-*.f64 66.2

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

7. Applied rewrites 66.2%

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

8. Final simplification 66.2%

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

Alternative 12: 65.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

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

5. Applied rewrites 66.2%

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

Alternative 13: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 39.0

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

5. Applied rewrites 39.0%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. lower-/.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 39.1

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

7. Applied rewrites 39.1%

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

Alternative 14: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in y around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 39.0

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

5. Applied rewrites 39.0%

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

Developer Target 1: 96.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))