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

Percentage Accurate: 95.3% → 97.1%

Time: 11.0s

Alternatives: 17

Speedup: 1.0×

• 29.0% 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: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0)))) 2e+277)
   (- (fma (/ y z) -0.3333333333333333 x) (/ t (* (* y z) -3.0)))
   (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+277) {
		tmp = fma((y / z), -0.3333333333333333, x) - (t / ((y * z) * -3.0));
	} else {
		tmp = fma((0.3333333333333333 / z), ((t / y) - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0)))) <= 2e+277)
		tmp = Float64(fma(Float64(y / z), -0.3333333333333333, x) - Float64(t / Float64(Float64(y * z) * -3.0)));
	else
		tmp = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+277], N[(N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] - N[(t / N[(N[(y * z), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.00000000000000001e277

   1. Initial program 98.7%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if 2.00000000000000001e277 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0)))) 5e+303)
   (fma (/ t (* y z)) 0.3333333333333333 (fma (/ y z) -0.3333333333333333 x))
   (+ x (/ (- (/ t y) y) (* z 3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 5e+303) {
		tmp = fma((t / (y * z)), 0.3333333333333333, fma((y / z), -0.3333333333333333, x));
	} else {
		tmp = x + (((t / y) - y) / (z * 3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0)))) <= 5e+303)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, fma(Float64(y / z), -0.3333333333333333, x));
	else
		tmp = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+303], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 4.9999999999999997e303

   1. Initial program 98.7%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. 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. *-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 98.7

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/r* N/A

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

      20. div-inv N/A

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

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

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

      22. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if 4.9999999999999997e303 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 84.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.3333333333333333, \frac{1}{z}, x\right)\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.1e+49)
   (fma (* y -0.3333333333333333) (/ 1.0 z) x)
   (if (<= y -1.15e-23)
     (* (/ 0.3333333333333333 z) (- (/ t y) y))
     (if (<= y 1.35e+56)
       (fma (/ t z) (/ 0.3333333333333333 y) x)
       (fma y (/ -0.3333333333333333 z) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.1e+49) {
		tmp = fma((y * -0.3333333333333333), (1.0 / z), x);
	} else if (y <= -1.15e-23) {
		tmp = (0.3333333333333333 / z) * ((t / y) - y);
	} else if (y <= 1.35e+56) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.1e+49)
		tmp = fma(Float64(y * -0.3333333333333333), Float64(1.0 / z), x);
	elseif (y <= -1.15e-23)
		tmp = Float64(Float64(0.3333333333333333 / z) * Float64(Float64(t / y) - y));
	elseif (y <= 1.35e+56)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.1e+49], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, -1.15e-23], N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+56], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.09999999999999992e49

   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

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

      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. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -3.09999999999999992e49 < y < -1.15000000000000005e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -1.15000000000000005e-23 < y < 1.35000000000000005e56

   1. Initial program 92.8%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 94.8%

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

   if 1.35000000000000005e56 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.3333333333333333, \frac{1}{z}, x\right)\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (- (/ t y) y) (* z 3.0)))))
   (if (<= y -2.3e-63)
     t_1
     (if (<= y 3.4e-37) (fma (/ t z) (/ 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((t / y) - y) / (z * 3.0));
	double tmp;
	if (y <= -2.3e-63) {
		tmp = t_1;
	} else if (y <= 3.4e-37) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)))
	tmp = 0.0
	if (y <= -2.3e-63)
		tmp = t_1;
	elseif (y <= 3.4e-37)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-63], t$95$1, If[LessEqual[y, 3.4e-37], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e-63 or 3.40000000000000018e-37 < y

   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

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -2.3e-63 < y < 3.40000000000000018e-37

   1. Initial program 91.1%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 89.5

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 97.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-63}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)))
   (if (<= y -2.3e-63)
     t_1
     (if (<= y 3.4e-37) (fma (/ t z) (/ 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((0.3333333333333333 / z), ((t / y) - y), x);
	double tmp;
	if (y <= -2.3e-63) {
		tmp = t_1;
	} else if (y <= 3.4e-37) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x)
	tmp = 0.0
	if (y <= -2.3e-63)
		tmp = t_1;
	elseif (y <= 3.4e-37)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.3e-63], t$95$1, If[LessEqual[y, 3.4e-37], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e-63 or 3.40000000000000018e-37 < y

   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. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -2.3e-63 < y < 3.40000000000000018e-37

   1. Initial program 91.1%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 89.5

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 97.1%

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

Alternative 6: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.3333333333333333, \frac{1}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.4e-23)
   (fma (* y -0.3333333333333333) (/ 1.0 z) x)
   (if (<= y 1.35e+56)
     (fma (/ t z) (/ 0.3333333333333333 y) x)
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.4e-23) {
		tmp = fma((y * -0.3333333333333333), (1.0 / z), x);
	} else if (y <= 1.35e+56) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.4e-23)
		tmp = fma(Float64(y * -0.3333333333333333), Float64(1.0 / z), x);
	elseif (y <= 1.35e+56)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.4e-23], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.35e+56], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4000000000000001e-23

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 89.8

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

   9. Applied rewrites 89.8%

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

   if -3.4000000000000001e-23 < y < 1.35000000000000005e56

   1. Initial program 92.8%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 94.8%

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

   if 1.35000000000000005e56 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.3333333333333333, \frac{1}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.3333333333333333, \frac{1}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.4e-23)
   (fma (* y -0.3333333333333333) (/ 1.0 z) x)
   (if (<= y 1.35e+56)
     (fma 0.3333333333333333 (/ t (* y z)) x)
     (fma y (/ -0.3333333333333333 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.4e-23) {
		tmp = fma((y * -0.3333333333333333), (1.0 / z), x);
	} else if (y <= 1.35e+56) {
		tmp = fma(0.3333333333333333, (t / (y * z)), x);
	} else {
		tmp = fma(y, (-0.3333333333333333 / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.4e-23)
		tmp = fma(Float64(y * -0.3333333333333333), Float64(1.0 / z), x);
	elseif (y <= 1.35e+56)
		tmp = fma(0.3333333333333333, Float64(t / Float64(y * z)), x);
	else
		tmp = fma(y, Float64(-0.3333333333333333 / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.4e-23], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.35e+56], N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4000000000000001e-23

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 89.8

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

   9. Applied rewrites 89.8%

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

   if -3.4000000000000001e-23 < y < 1.35000000000000005e56

   1. Initial program 92.8%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if 1.35000000000000005e56 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.3333333333333333, \frac{1}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -3.4e-23)
     t_1
     (if (<= y 1.35e+56) (fma 0.3333333333333333 (/ t (* y z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -3.4e-23) {
		tmp = t_1;
	} else if (y <= 1.35e+56) {
		tmp = fma(0.3333333333333333, (t / (y * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -3.4e-23)
		tmp = t_1;
	elseif (y <= 1.35e+56)
		tmp = fma(0.3333333333333333, Float64(t / Float64(y * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -3.4e-23], t$95$1, If[LessEqual[y, 1.35e+56], N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4000000000000001e-23 or 1.35000000000000005e56 < y

   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. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 93.3%

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

   if -3.4000000000000001e-23 < y < 1.35000000000000005e56

   1. Initial program 92.8%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

Alternative 9: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1.85e-53)
     t_1
     (if (<= y 1.65e-30) (/ (* t 0.3333333333333333) (* y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1.85e-53) {
		tmp = t_1;
	} else if (y <= 1.65e-30) {
		tmp = (t * 0.3333333333333333) / (y * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1.85e-53)
		tmp = t_1;
	elseif (y <= 1.65e-30)
		tmp = Float64(Float64(t * 0.3333333333333333) / Float64(y * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.85e-53], t$95$1, If[LessEqual[y, 1.65e-30], N[(N[(t * 0.3333333333333333), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999991e-53 or 1.6500000000000001e-30 < y

   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. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 88.9%

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

   if -1.84999999999999991e-53 < y < 1.6500000000000001e-30

   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. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{t \cdot 0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1.85e-53) t_1 (if (<= y 1.65e-30) (/ t (* 3.0 (* y z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1.85e-53) {
		tmp = t_1;
	} else if (y <= 1.65e-30) {
		tmp = t / (3.0 * (y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1.85e-53)
		tmp = t_1;
	elseif (y <= 1.65e-30)
		tmp = Float64(t / Float64(3.0 * Float64(y * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.85e-53], t$95$1, If[LessEqual[y, 1.65e-30], N[(t / N[(3.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999991e-53 or 1.6500000000000001e-30 < y

   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. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 88.9%

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

   if -1.84999999999999991e-53 < y < 1.6500000000000001e-30

   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. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.3%

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

   9. Applied rewrites 71.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1.85e-53) t_1 (if (<= y 1.65e-30) (/ t (* y (* z 3.0))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1.85e-53) {
		tmp = t_1;
	} else if (y <= 1.65e-30) {
		tmp = t / (y * (z * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1.85e-53)
		tmp = t_1;
	elseif (y <= 1.65e-30)
		tmp = Float64(t / Float64(y * Float64(z * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.85e-53], t$95$1, If[LessEqual[y, 1.65e-30], N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999991e-53 or 1.6500000000000001e-30 < y

   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. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 88.9%

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

   if -1.84999999999999991e-53 < y < 1.6500000000000001e-30

   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. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.3%

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

Alternative 12: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1.85e-53)
     t_1
     (if (<= y 1.65e-30) (* 0.3333333333333333 (/ t (* y z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1.85e-53) {
		tmp = t_1;
	} else if (y <= 1.65e-30) {
		tmp = 0.3333333333333333 * (t / (y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1.85e-53)
		tmp = t_1;
	elseif (y <= 1.65e-30)
		tmp = Float64(0.3333333333333333 * Float64(t / Float64(y * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.85e-53], t$95$1, If[LessEqual[y, 1.65e-30], N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999991e-53 or 1.6500000000000001e-30 < y

   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. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 88.9%

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

   if -1.84999999999999991e-53 < y < 1.6500000000000001e-30

   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. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-30}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1.85e-53)
     t_1
     (if (<= y 1.6e-30) (* t (/ 0.3333333333333333 (* y z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1.85e-53) {
		tmp = t_1;
	} else if (y <= 1.6e-30) {
		tmp = t * (0.3333333333333333 / (y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1.85e-53)
		tmp = t_1;
	elseif (y <= 1.6e-30)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(y * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.85e-53], t$95$1, If[LessEqual[y, 1.6e-30], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999991e-53 or 1.6e-30 < y

   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. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 88.9%

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

   if -1.84999999999999991e-53 < y < 1.6e-30

   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. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.3%

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

Alternative 14: 63.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

5. Applied rewrites 59.4%

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

Alternative 15: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * -0.3333333333333333), $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. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 35.9

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

5. Applied rewrites 35.9%

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

7. Applied rewrites 35.9%

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

Alternative 16: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 35.9

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

5. Applied rewrites 35.9%

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

7. Applied rewrites 35.9%

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

Alternative 17: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 35.9

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

5. Applied rewrites 35.9%

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

Developer Target 1: 96.0% 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 2024226 
(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))))