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

Percentage Accurate: 96.0% → 98.1%

Time: 7.1s

Alternatives: 11

Speedup: 1.0×

• 27.9% 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: 96.0% 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: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-81}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(3 \cdot y\right) \cdot z}\\ \mathbf{elif}\;y \leq 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6e-81)
   (+ (- x (/ y (* z 3.0))) (/ t (* (* 3.0 y) z)))
   (if (<= y 1e-127)
     (fma (/ 0.3333333333333333 y) (/ t z) x)
     (- x (/ (- y (/ t y)) (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6e-81) {
		tmp = (x - (y / (z * 3.0))) + (t / ((3.0 * y) * z));
	} else if (y <= 1e-127) {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6e-81)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(3.0 * y) * z)));
	elseif (y <= 1e-127)
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6e-81], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(3.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-127], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.9999999999999998e-81

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -5.9999999999999998e-81 < y < 1e-127

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1e-127 < y

   1. Initial program 96.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;y \leq 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6e-81)
   (+ (fma -0.3333333333333333 (/ y z) x) (/ t (* (* z 3.0) y)))
   (if (<= y 1e-127)
     (fma (/ 0.3333333333333333 y) (/ t z) x)
     (- x (/ (- y (/ t y)) (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6e-81) {
		tmp = fma(-0.3333333333333333, (y / z), x) + (t / ((z * 3.0) * y));
	} else if (y <= 1e-127) {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	} else {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6e-81)
		tmp = Float64(fma(-0.3333333333333333, Float64(y / z), x) + Float64(t / Float64(Float64(z * 3.0) * y)));
	elseif (y <= 1e-127)
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	else
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6e-81], N[(N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-127], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.9999999999999998e-81

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -5.9999999999999998e-81 < y < 1e-127

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 1e-127 < y

   1. Initial program 96.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-81} \lor \neg \left(y \leq 10^{-127}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -7e-81) (not (<= y 1e-127)))
   (- x (/ (- y (/ t y)) (* 3.0 z)))
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -7e-81) || !(y <= 1e-127)) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -7e-81) || !(y <= 1e-127))
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -7e-81], N[Not[LessEqual[y, 1e-127]], $MachinePrecision]], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999973e-81 or 1e-127 < y

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.3

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if -6.99999999999999973e-81 < y < 1e-127

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-81} \lor \neg \left(y \leq 10^{-127}\right):\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+81} \lor \neg \left(y \leq 4.1 \cdot 10^{+43}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e+81) (not (<= y 4.1e+43)))
   (- x (/ (/ y z) 3.0))
   (fma (/ 0.3333333333333333 y) (/ t z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e+81) || !(y <= 4.1e+43)) {
		tmp = x - ((y / z) / 3.0);
	} else {
		tmp = fma((0.3333333333333333 / y), (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e+81) || !(y <= 4.1e+43))
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	else
		tmp = fma(Float64(0.3333333333333333 / y), Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e+81], N[Not[LessEqual[y, 4.1e+43]], $MachinePrecision]], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / y), $MachinePrecision] * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7999999999999999e81 or 4.1e43 < y

   1. Initial program 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate--r- N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift--.f64 99.8

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 97.9

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

   9. Applied rewrites 97.9%

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

   if -5.7999999999999999e81 < y < 4.1e43

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+81} \lor \neg \left(y \leq 4.1 \cdot 10^{+43}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{y}, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+81} \lor \neg \left(y \leq 1.45 \cdot 10^{+33}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e+81) (not (<= y 1.45e+33)))
   (- x (/ (/ y z) 3.0))
   (fma 0.3333333333333333 (/ t (* z y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e+81) || !(y <= 1.45e+33)) {
		tmp = x - ((y / z) / 3.0);
	} else {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e+81) || !(y <= 1.45e+33))
		tmp = Float64(x - Float64(Float64(y / z) / 3.0));
	else
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e+81], N[Not[LessEqual[y, 1.45e+33]], $MachinePrecision]], N[(x - N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7999999999999999e81 or 1.45000000000000012e33 < y

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. associate--r- N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 96.9

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

   9. Applied rewrites 96.9%

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

   if -5.7999999999999999e81 < y < 1.45000000000000012e33

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 86.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+81} \lor \neg \left(y \leq 1.45 \cdot 10^{+33}\right):\\ \;\;\;\;x - \frac{\frac{y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+81} \lor \neg \left(y \leq 1.45 \cdot 10^{+33}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.8e+81) (not (<= y 1.45e+33)))
   (fma -0.3333333333333333 (/ y z) x)
   (fma 0.3333333333333333 (/ t (* z y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.8e+81) || !(y <= 1.45e+33)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = fma(0.3333333333333333, (t / (z * y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.8e+81) || !(y <= 1.45e+33))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = fma(0.3333333333333333, Float64(t / Float64(z * y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.8e+81], N[Not[LessEqual[y, 1.45e+33]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.7999999999999999e81 or 1.45000000000000012e33 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. distribute-lft-neg-in 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. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

      20. fp-cancel-sign-sub-inv N/A

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -5.7999999999999999e81 < y < 1.45000000000000012e33

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. div-add N/A

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

      2. associate-/l* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r/ N/A

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

      14. times-frac N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 86.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+81} \lor \neg \left(y \leq 1.45 \cdot 10^{+33}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{z \cdot y}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-48} \lor \neg \left(y \leq 1.16 \cdot 10^{-112}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.3e-48) (not (<= y 1.16e-112)))
   (fma -0.3333333333333333 (/ y z) x)
   (* (/ t (* z y)) 0.3333333333333333)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.3e-48) || !(y <= 1.16e-112)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = (t / (z * y)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.3e-48) || !(y <= 1.16e-112))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.3e-48], N[Not[LessEqual[y, 1.16e-112]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3000000000000001e-48 or 1.16000000000000002e-112 < y

   1. Initial program 98.0%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. distribute-lft-neg-in 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. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

      20. fp-cancel-sign-sub-inv N/A

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -2.3000000000000001e-48 < y < 1.16000000000000002e-112

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-48} \lor \neg \left(y \leq 1.16 \cdot 10^{-112}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-84} \lor \neg \left(y \leq 1.16 \cdot 10^{-112}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.6e-84) (not (<= y 1.16e-112)))
   (fma -0.3333333333333333 (/ y z) x)
   (* t (/ 0.3333333333333333 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.6e-84) || !(y <= 1.16e-112)) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else {
		tmp = t * (0.3333333333333333 / (z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.6e-84) || !(y <= 1.16e-112))
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	else
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(z * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.6e-84], N[Not[LessEqual[y, 1.16e-112]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e-84 or 1.16000000000000002e-112 < y

   1. Initial program 98.1%

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

   3. Taylor expanded in y around inf

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

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. distribute-lft-neg-in 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. *-rgt-identity N/A

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

      18. *-lft-identity N/A

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

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

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

      20. fp-cancel-sign-sub-inv N/A

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

      21. *-lft-identity N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -1.6e-84 < y < 1.16000000000000002e-112

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 69.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-84} \lor \neg \left(y \leq 1.16 \cdot 10^{-112}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

3. Taylor expanded in y around inf

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

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

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

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

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

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

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

   12. associate-/l* N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. distribute-lft-neg-in 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. *-rgt-identity N/A

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

   18. *-lft-identity N/A

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

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

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

   20. fp-cancel-sign-sub-inv N/A

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

   21. *-lft-identity N/A

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

   22. lower-fma.f64 N/A

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

   23. lower-/.f64 62.2

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

5. Applied rewrites 62.2%

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

Alternative 10: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 95.4

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

4. Applied rewrites 95.4%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f64 N/A

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

   12. associate--r- N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. div-sub N/A

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

   18. lift--.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. lift--.f64 93.6

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

6. Applied rewrites 93.6%

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

7. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 32.3

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

9. Applied rewrites 32.3%

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

10. Final simplification 32.3%

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

Alternative 11: 36.1% 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 95.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 95.4

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

4. Applied rewrites 95.4%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/r* N/A

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

   11. lift-*.f64 N/A

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

   12. associate--r- N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. div-sub N/A

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

   18. lift--.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. lift--.f64 93.6

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

6. Applied rewrites 93.6%

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

7. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 32.3

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

9. Applied rewrites 32.3%

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

11. Applied rewrites 32.3%

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

12. Final simplification 32.3%

    \[\leadsto y \cdot \frac{-0.3333333333333333}{z} \]
13. 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 2024338 
(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))))