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

Percentage Accurate: 95.3% → 99.5%

Time: 10.1s

Alternatives: 19

Speedup: 1.4×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.3% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -4 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{z}}{y}, t, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)} - \left(\frac{\frac{y}{z}}{3} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* 3.0 z) -4e+53)
   (fma (/ (/ 0.3333333333333333 z) y) t (fma -0.3333333333333333 (/ y z) x))
   (if (<= (* 3.0 z) 1e-63)
     (- x (/ (- y (/ t y)) (* 3.0 z)))
     (- (/ t (* y (* 3.0 z))) (- (/ (/ y z) 3.0) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((3.0 * z) <= -4e+53) {
		tmp = fma(((0.3333333333333333 / z) / y), t, fma(-0.3333333333333333, (y / z), x));
	} else if ((3.0 * z) <= 1e-63) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = (t / (y * (3.0 * z))) - (((y / z) / 3.0) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(3.0 * z) <= -4e+53)
		tmp = fma(Float64(Float64(0.3333333333333333 / z) / y), t, fma(-0.3333333333333333, Float64(y / z), x));
	elseif (Float64(3.0 * z) <= 1e-63)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = Float64(Float64(t / Float64(y * Float64(3.0 * z))) - Float64(Float64(Float64(y / z) / 3.0) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(3.0 * z), $MachinePrecision], -4e+53], N[(N[(N[(0.3333333333333333 / z), $MachinePrecision] / y), $MachinePrecision] * t + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * z), $MachinePrecision], 1e-63], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(y * N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y / z), $MachinePrecision] / 3.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -4e53

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. distribute-neg-frac N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. times-frac N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

   if -4e53 < (*.f64 z #s(literal 3 binary64)) < 1.00000000000000007e-63

   1. Initial program 91.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.00000000000000007e-63 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -4 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{z}}{y}, t, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)} - \left(\frac{\frac{y}{z}}{3} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -4 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{z}}{y}, t, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)} - \left(\frac{y}{3 \cdot z} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* 3.0 z) -4e+53)
   (fma (/ (/ 0.3333333333333333 z) y) t (fma -0.3333333333333333 (/ y z) x))
   (if (<= (* 3.0 z) 1e-63)
     (- x (/ (- y (/ t y)) (* 3.0 z)))
     (- (/ t (* y (* 3.0 z))) (- (/ y (* 3.0 z)) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((3.0 * z) <= -4e+53) {
		tmp = fma(((0.3333333333333333 / z) / y), t, fma(-0.3333333333333333, (y / z), x));
	} else if ((3.0 * z) <= 1e-63) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = (t / (y * (3.0 * z))) - ((y / (3.0 * z)) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(3.0 * z) <= -4e+53)
		tmp = fma(Float64(Float64(0.3333333333333333 / z) / y), t, fma(-0.3333333333333333, Float64(y / z), x));
	elseif (Float64(3.0 * z) <= 1e-63)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = Float64(Float64(t / Float64(y * Float64(3.0 * z))) - Float64(Float64(y / Float64(3.0 * z)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(3.0 * z), $MachinePrecision], -4e+53], N[(N[(N[(0.3333333333333333 / z), $MachinePrecision] / y), $MachinePrecision] * t + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * z), $MachinePrecision], 1e-63], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(y * N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -4e53

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval 99.9

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

      15. lift--.f64 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. distribute-neg-frac N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. times-frac N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

   if -4e53 < (*.f64 z #s(literal 3 binary64)) < 1.00000000000000007e-63

   1. Initial program 91.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.00000000000000007e-63 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -4 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{0.3333333333333333}{z}}{y}, t, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)} - \left(\frac{y}{3 \cdot z} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{y \cdot \left(3 \cdot z\right)}\\ \mathbf{if}\;3 \cdot z \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, t\_1 + x\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \left(\frac{y}{3 \cdot z} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* y (* 3.0 z)))))
   (if (<= (* 3.0 z) -5e-26)
     (fma (/ -0.3333333333333333 z) y (+ t_1 x))
     (if (<= (* 3.0 z) 1e-63)
       (- x (/ (- y (/ t y)) (* 3.0 z)))
       (- t_1 (- (/ y (* 3.0 z)) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t / (y * (3.0 * z));
	double tmp;
	if ((3.0 * z) <= -5e-26) {
		tmp = fma((-0.3333333333333333 / z), y, (t_1 + x));
	} else if ((3.0 * z) <= 1e-63) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = t_1 - ((y / (3.0 * z)) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t / Float64(y * Float64(3.0 * z)))
	tmp = 0.0
	if (Float64(3.0 * z) <= -5e-26)
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(t_1 + x));
	elseif (Float64(3.0 * z) <= 1e-63)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = Float64(t_1 - Float64(Float64(y / Float64(3.0 * z)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(y * N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(3.0 * z), $MachinePrecision], -5e-26], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(t$95$1 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * z), $MachinePrecision], 1e-63], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -5.00000000000000019e-26

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 99.9

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -5.00000000000000019e-26 < (*.f64 z #s(literal 3 binary64)) < 1.00000000000000007e-63

   1. Initial program 90.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. 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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.00000000000000007e-63 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{y \cdot \left(3 \cdot z\right)} + x\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)} - \left(\frac{y}{3 \cdot z} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{y \cdot \left(3 \cdot z\right)} + x\right)\\ \mathbf{if}\;3 \cdot z \leq -5 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ -0.3333333333333333 z) y (+ (/ t (* y (* 3.0 z))) x))))
   (if (<= (* 3.0 z) -5e-26)
     t_1
     (if (<= (* 3.0 z) 1e-63) (- x (/ (- y (/ t y)) (* 3.0 z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((-0.3333333333333333 / z), y, ((t / (y * (3.0 * z))) + x));
	double tmp;
	if ((3.0 * z) <= -5e-26) {
		tmp = t_1;
	} else if ((3.0 * z) <= 1e-63) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(-0.3333333333333333 / z), y, Float64(Float64(t / Float64(y * Float64(3.0 * z))) + x))
	tmp = 0.0
	if (Float64(3.0 * z) <= -5e-26)
		tmp = t_1;
	elseif (Float64(3.0 * z) <= 1e-63)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(N[(t / N[(y * N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(3.0 * z), $MachinePrecision], -5e-26], t$95$1, If[LessEqual[N[(3.0 * z), $MachinePrecision], 1e-63], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -5.00000000000000019e-26 or 1.00000000000000007e-63 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 99.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -5.00000000000000019e-26 < (*.f64 z #s(literal 3 binary64)) < 1.00000000000000007e-63

   1. Initial program 90.3%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. 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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{y \cdot \left(3 \cdot z\right)} + x\right)\\ \mathbf{elif}\;3 \cdot z \leq 10^{-63}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{y \cdot \left(3 \cdot z\right)} + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (- y (/ t y)) (* 3.0 z)))))
   (if (<= y -4.8e-41)
     t_1
     (if (<= y 3.2e-125) (/ (fma (/ t z) 0.3333333333333333 (* x y)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y - (t / y)) / (3.0 * z));
	double tmp;
	if (y <= -4.8e-41) {
		tmp = t_1;
	} else if (y <= 3.2e-125) {
		tmp = fma((t / z), 0.3333333333333333, (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -4.8e-41)
		tmp = t_1;
	elseif (y <= 3.2e-125)
		tmp = Float64(fma(Float64(t / z), 0.3333333333333333, Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e-41], t$95$1, If[LessEqual[y, 3.2e-125], N[(N[(N[(t / z), $MachinePrecision] * 0.3333333333333333 + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000044e-41 or 3.1999999999999998e-125 < 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

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

      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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   if -4.80000000000000044e-41 < y < 3.1999999999999998e-125

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ (- y (/ t y)) z) -0.3333333333333333 x)))
   (if (<= y -4.8e-41)
     t_1
     (if (<= y 1.15e-155)
       (/ (fma (/ t z) 0.3333333333333333 (* x y)) y)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((y - (t / y)) / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -4.8e-41) {
		tmp = t_1;
	} else if (y <= 1.15e-155) {
		tmp = fma((t / z), 0.3333333333333333, (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(y - Float64(t / y)) / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -4.8e-41)
		tmp = t_1;
	elseif (y <= 1.15e-155)
		tmp = Float64(fma(Float64(t / z), 0.3333333333333333, Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -4.8e-41], t$95$1, If[LessEqual[y, 1.15e-155], N[(N[(N[(t / z), $MachinePrecision] * 0.3333333333333333 + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000044e-41 or 1.15000000000000003e-155 < 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 t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 98.6%

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

   if -4.80000000000000044e-41 < y < 1.15000000000000003e-155

   1. Initial program 92.9%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -175000000000:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 3}{\left(3 \cdot z\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -175000000000.0)
   (- x (* (/ y z) 0.3333333333333333))
   (if (<= y 2.1e+65)
     (fma (/ t (* y z)) 0.3333333333333333 x)
     (- x (/ (* y 3.0) (* (* 3.0 z) 3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -175000000000.0) {
		tmp = x - ((y / z) * 0.3333333333333333);
	} else if (y <= 2.1e+65) {
		tmp = fma((t / (y * z)), 0.3333333333333333, x);
	} else {
		tmp = x - ((y * 3.0) / ((3.0 * z) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -175000000000.0)
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	elseif (y <= 2.1e+65)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(Float64(y * 3.0) / Float64(Float64(3.0 * z) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -175000000000.0], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+65], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(N[(y * 3.0), $MachinePrecision] / N[(N[(3.0 * z), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75e11

   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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -1.75e11 < y < 2.09999999999999991e65

   1. Initial program 94.4%

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

      1. lift-+.f64 N/A

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

      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 89.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

   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

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

      8. associate-/l/ N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. remove-double-neg N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

   7. Applied rewrites 88.5%

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

   if 2.09999999999999991e65 < 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

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 95.0

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

   9. Applied rewrites 95.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -175000000000:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 3}{\left(3 \cdot z\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -175000000000:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -175000000000.0)
   (- x (* (/ y z) 0.3333333333333333))
   (if (<= y 2.1e+65)
     (fma (/ t (* y z)) 0.3333333333333333 x)
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -175000000000.0) {
		tmp = x - ((y / z) * 0.3333333333333333);
	} else if (y <= 2.1e+65) {
		tmp = fma((t / (y * z)), 0.3333333333333333, x);
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -175000000000.0)
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	elseif (y <= 2.1e+65)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, x);
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -175000000000.0], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+65], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75e11

   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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -1.75e11 < y < 2.09999999999999991e65

   1. Initial program 94.4%

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

      1. lift-+.f64 N/A

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

      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 89.2

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.2

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

   4. Applied rewrites 89.2%

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

   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

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

      8. associate-/l/ N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. remove-double-neg N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

   7. Applied rewrites 88.5%

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

   if 2.09999999999999991e65 < 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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -175000000000:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{t}{\left(y \cdot 3\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.15e-19)
   (- x (* (/ y z) 0.3333333333333333))
   (if (<= y 7.2e-200)
     (/ t (* (* y 3.0) z))
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.15e-19) {
		tmp = x - ((y / z) * 0.3333333333333333);
	} else if (y <= 7.2e-200) {
		tmp = t / ((y * 3.0) * z);
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.15e-19)
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	elseif (y <= 7.2e-200)
		tmp = Float64(t / Float64(Float64(y * 3.0) * z));
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.15e-19], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-200], N[(t / N[(N[(y * 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15000000000000009e-19

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -3.15000000000000009e-19 < y < 7.2000000000000003e-200

   1. Initial program 92.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.6%

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

   9. Applied rewrites 64.8%

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

   11. Applied rewrites 64.8%

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

   if 7.2000000000000003e-200 < 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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{t}{\left(y \cdot 3\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.15e-19)
   (- x (* (/ y z) 0.3333333333333333))
   (if (<= y 7.2e-200)
     (/ t (* y (* 3.0 z)))
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.15e-19) {
		tmp = x - ((y / z) * 0.3333333333333333);
	} else if (y <= 7.2e-200) {
		tmp = t / (y * (3.0 * z));
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.15e-19)
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	elseif (y <= 7.2e-200)
		tmp = Float64(t / Float64(y * Float64(3.0 * z)));
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.15e-19], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-200], N[(t / N[(y * N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15000000000000009e-19

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -3.15000000000000009e-19 < y < 7.2000000000000003e-200

   1. Initial program 92.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

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

   if 7.2000000000000003e-200 < 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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{t}{y \cdot \left(3 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.15e-19)
   (- x (* (/ y z) 0.3333333333333333))
   (if (<= y 7.2e-200)
     (* (/ t (* y z)) 0.3333333333333333)
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.15e-19) {
		tmp = x - ((y / z) * 0.3333333333333333);
	} else if (y <= 7.2e-200) {
		tmp = (t / (y * z)) * 0.3333333333333333;
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.15e-19)
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	elseif (y <= 7.2e-200)
		tmp = Float64(Float64(t / Float64(y * z)) * 0.3333333333333333);
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.15e-19], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-200], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15000000000000009e-19

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -3.15000000000000009e-19 < y < 7.2000000000000003e-200

   1. Initial program 92.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   if 7.2000000000000003e-200 < 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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{0.3333333333333333}{y \cdot z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.15e-19)
   (- x (* (/ y z) 0.3333333333333333))
   (if (<= y 7.2e-200)
     (* (/ 0.3333333333333333 (* y z)) t)
     (fma -0.3333333333333333 (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.15e-19) {
		tmp = x - ((y / z) * 0.3333333333333333);
	} else if (y <= 7.2e-200) {
		tmp = (0.3333333333333333 / (y * z)) * t;
	} else {
		tmp = fma(-0.3333333333333333, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.15e-19)
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	elseif (y <= 7.2e-200)
		tmp = Float64(Float64(0.3333333333333333 / Float64(y * z)) * t);
	else
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.15e-19], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-200], N[(N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.15000000000000009e-19

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if -3.15000000000000009e-19 < y < 7.2000000000000003e-200

   1. Initial program 92.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.6%

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

   9. Applied rewrites 63.9%

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

   if 7.2000000000000003e-200 < 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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{0.3333333333333333}{y \cdot z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in t around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

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

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

   4. associate-/r* N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

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

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

   10. associate-*r/ N/A

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

5. Applied rewrites 93.5%

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

Alternative 14: 64.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

   1. lift-+.f64 N/A

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

   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 93.8

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

   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 93.8

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

4. Applied rewrites 93.8%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 65.5

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

7. Applied rewrites 65.5%

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

8. Final simplification 65.5%

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

Alternative 15: 64.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in t around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 65.5

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

5. Applied rewrites 65.5%

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

Alternative 16: 36.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 30.5

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

5. Applied rewrites 30.5%

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

7. Applied rewrites 30.5%

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

8. Final simplification 30.5%

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

Alternative 17: 36.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 30.5

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

5. Applied rewrites 30.5%

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

7. Applied rewrites 30.5%

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

Alternative 18: 36.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 30.5

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

5. Applied rewrites 30.5%

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

7. Applied rewrites 30.5%

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

Alternative 19: 36.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 30.5

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

5. Applied rewrites 30.5%

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

6. Final simplification 30.5%

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

Developer Target 1: 95.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024244 
(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))))