Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.9% → 99.9%

Time: 7.2s

Alternatives: 11

Speedup: 1.2×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma y 5.0 (* (fma (+ z y) 2.0 t) x)))

C

double code(double x, double y, double z, double t) {
	return fma(y, 5.0, (fma((z + y), 2.0, t) * x));
}

Julia

function code(x, y, z, t)
	return fma(y, 5.0, Float64(fma(Float64(z + y), 2.0, t) * x))
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(x + \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto t \cdot \color{blue}{\left(\left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + x\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + t \cdot x} \]

   3. remove-double-neg N/A

      \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t \cdot x\right)\right)\right)\right)} \]

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

      \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) - t \cdot \left(-1 \cdot x\right)} \]

   7. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(5 \cdot \frac{y}{t}\right) \cdot t + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t\right)} - t \cdot \left(-1 \cdot x\right) \]

   8. associate-+r- N/A

      \[\leadsto \color{blue}{\left(5 \cdot \frac{y}{t}\right) \cdot t + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right)} \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t}\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

   10. *-commutative N/A

       \[\leadsto t \cdot \color{blue}{\left(\frac{y}{t} \cdot 5\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \color{blue}{\left(t \cdot \frac{y}{t}\right) \cdot 5} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(t \cdot -1\right) \cdot x}\right) \]

   13. *-commutative N/A

       \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(-1 \cdot t\right)} \cdot x\right) \]

   14. mul-1-neg N/A

       \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot x\right) \]

5. Applied rewrites 90.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot \frac{y}{t}, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(y \cdot 1, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right) \]

8. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right) \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -160000 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -160000.0) (not (<= x 2.5)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* (fma 2.0 z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -160000.0) || !(x <= 2.5)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, (fma(2.0, z, t) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -160000.0) || !(x <= 2.5))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(fma(2.0, z, t) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -160000.0], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e5 or 2.5 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(x + \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + x\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + t \cdot x} \]

      3. remove-double-neg N/A

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t \cdot x\right)\right)\right)\right)} \]

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

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) - t \cdot \left(-1 \cdot x\right)} \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(5 \cdot \frac{y}{t}\right) \cdot t + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t\right)} - t \cdot \left(-1 \cdot x\right) \]

      8. associate-+r- N/A

         \[\leadsto \color{blue}{\left(5 \cdot \frac{y}{t}\right) \cdot t + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t}\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto t \cdot \color{blue}{\left(\frac{y}{t} \cdot 5\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(t \cdot \frac{y}{t}\right) \cdot 5} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(t \cdot -1\right) \cdot x}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(-1 \cdot t\right)} \cdot x\right) \]

      14. mul-1-neg N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot x\right) \]

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot \frac{y}{t}, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -1.6e5 < x < 2.5

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 99.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

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

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

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

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.6

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -160000 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -2.15e+79)
     t_1
     (if (<= y 1.5e-106)
       (* (fma 2.0 z t) x)
       (if (<= y 1.3e+165) (fma y 5.0 (* x t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, x, 5.0) * y;
	double tmp;
	if (y <= -2.15e+79) {
		tmp = t_1;
	} else if (y <= 1.5e-106) {
		tmp = fma(2.0, z, t) * x;
	} else if (y <= 1.3e+165) {
		tmp = fma(y, 5.0, (x * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, x, 5.0) * y)
	tmp = 0.0
	if (y <= -2.15e+79)
		tmp = t_1;
	elseif (y <= 1.5e-106)
		tmp = Float64(fma(2.0, z, t) * x);
	elseif (y <= 1.3e+165)
		tmp = fma(y, 5.0, Float64(x * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.15e+79], t$95$1, If[LessEqual[y, 1.5e-106], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.3e+165], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1500000000000002e79 or 1.3000000000000001e165 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      9. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \cdot y \]

      10. associate--r- N/A

          \[\leadsto \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \cdot y \]

      11. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(\color{blue}{2} \cdot x + 5\right) \cdot y \]

      14. lower-fma.f64 88.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -2.1500000000000002e79 < y < 1.50000000000000009e-106

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot z + t\right)} \cdot x \]

      4. lower-fma.f64 84.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, t\right) \cdot x} \]

   if 1.50000000000000009e-106 < y < 1.3000000000000001e165

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

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

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

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

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 87.3

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t \cdot x}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

      2. lower-*.f64 69.5

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-31} \lor \neg \left(x \leq 7.6 \cdot 10^{-60}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.2e-31) (not (<= x 7.6e-60)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* (* 2.0 z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e-31) || !(x <= 7.6e-60)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, ((2.0 * z) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.2e-31) || !(x <= 7.6e-60))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(Float64(2.0 * z) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.2e-31], N[Not[LessEqual[x, 7.6e-60]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(N[(2.0 * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999991e-31 or 7.59999999999999989e-60 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(x + \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + x\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + t \cdot x} \]

      3. remove-double-neg N/A

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t \cdot x\right)\right)\right)\right)} \]

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

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) - t \cdot \left(-1 \cdot x\right)} \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(5 \cdot \frac{y}{t}\right) \cdot t + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t\right)} - t \cdot \left(-1 \cdot x\right) \]

      8. associate-+r- N/A

         \[\leadsto \color{blue}{\left(5 \cdot \frac{y}{t}\right) \cdot t + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t}\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto t \cdot \color{blue}{\left(\frac{y}{t} \cdot 5\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(t \cdot \frac{y}{t}\right) \cdot 5} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(t \cdot -1\right) \cdot x}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(-1 \cdot t\right)} \cdot x\right) \]

      14. mul-1-neg N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot x\right) \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot \frac{y}{t}, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 98.2

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -5.19999999999999991e-31 < x < 7.59999999999999989e-60

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

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

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

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

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 80.8

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]

   7. Applied rewrites 80.8%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(2 \cdot z\right)} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-31} \lor \neg \left(x \leq 7.6 \cdot 10^{-60}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -160000 \lor \neg \left(x \leq 3.75 \cdot 10^{-100}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -160000.0) (not (<= x 3.75e-100)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -160000.0) || !(x <= 3.75e-100)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, (x * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -160000.0) || !(x <= 3.75e-100))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(x * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -160000.0], N[Not[LessEqual[x, 3.75e-100]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e5 or 3.75000000000000007e-100 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(x + \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto t \cdot \color{blue}{\left(\left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + x\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + t \cdot x} \]

      3. remove-double-neg N/A

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t \cdot x\right)\right)\right)\right)} \]

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

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) + \left(\mathsf{neg}\left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t} + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t}\right) - t \cdot \left(-1 \cdot x\right)} \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(5 \cdot \frac{y}{t}\right) \cdot t + \frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t\right)} - t \cdot \left(-1 \cdot x\right) \]

      8. associate-+r- N/A

         \[\leadsto \color{blue}{\left(5 \cdot \frac{y}{t}\right) \cdot t + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(5 \cdot \frac{y}{t}\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto t \cdot \color{blue}{\left(\frac{y}{t} \cdot 5\right)} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(t \cdot \frac{y}{t}\right) \cdot 5} + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - t \cdot \left(-1 \cdot x\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(t \cdot -1\right) \cdot x}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(-1 \cdot t\right)} \cdot x\right) \]

      14. mul-1-neg N/A

          \[\leadsto \left(t \cdot \frac{y}{t}\right) \cdot 5 + \left(\frac{x \cdot \left(2 \cdot y + 2 \cdot z\right)}{t} \cdot t - \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \cdot x\right) \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot \frac{y}{t}, 5, \mathsf{fma}\left(z + y, 2, t\right) \cdot x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + \left(2 \cdot y + 2 \cdot z\right)\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \cdot x \]

      4. distribute-lft-in N/A

         \[\leadsto \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \cdot x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

      7. lower-+.f64 97.6

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{z + y}, t\right) \cdot x \]

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, z + y, t\right) \cdot x} \]

   if -1.6e5 < x < 3.75000000000000007e-100

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot 5} + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \cdot x\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)} + t\right) \cdot x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\color{blue}{\left(\left(y + z\right) + z\right)} + y\right) + t\right) \cdot x\right) \]

      11. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right) \cdot x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right) \cdot x\right) \]

      14. flip-+ N/A

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

      15. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right) \cdot x\right) \]

      16. +-inverses N/A

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

      17. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right) \cdot x\right) \]

      18. +-inverses N/A

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

      19. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{\left(z + z\right)} + t\right) \cdot x\right) \]

      20. count-2 N/A

          \[\leadsto \mathsf{fma}\left(y, 5, \left(\color{blue}{2 \cdot z} + t\right) \cdot x\right) \]

      21. lower-fma.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{t \cdot x}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

      2. lower-*.f64 80.3

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

   7. Applied rewrites 80.3%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -160000 \lor \neg \left(x \leq 3.75 \cdot 10^{-100}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+79} \lor \neg \left(y \leq 7 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.15e+79) (not (<= y 7e+68)))
   (* (fma 2.0 x 5.0) y)
   (* (fma 2.0 z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.15e+79) || !(y <= 7e+68)) {
		tmp = fma(2.0, x, 5.0) * y;
	} else {
		tmp = fma(2.0, z, t) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.15e+79) || !(y <= 7e+68))
		tmp = Float64(fma(2.0, x, 5.0) * y);
	else
		tmp = Float64(fma(2.0, z, t) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.15e+79], N[Not[LessEqual[y, 7e+68]], $MachinePrecision]], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1500000000000002e79 or 6.99999999999999955e68 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      9. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \cdot y \]

      10. associate--r- N/A

          \[\leadsto \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \cdot y \]

      11. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(\color{blue}{2} \cdot x + 5\right) \cdot y \]

      14. lower-fma.f64 83.3

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   if -2.1500000000000002e79 < y < 6.99999999999999955e68

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot z\right) \cdot x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot z + t\right)} \cdot x \]

      4. lower-fma.f64 79.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, z, t\right)} \cdot x \]

   5. Applied rewrites 79.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+79} \lor \neg \left(y \leq 7 \cdot 10^{+68}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-31} \lor \neg \left(x \leq 3.4 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.2e-31) (not (<= x 3.4e-52))) (* (fma 2.0 y t) x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e-31) || !(x <= 3.4e-52)) {
		tmp = fma(2.0, y, t) * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.2e-31) || !(x <= 3.4e-52))
		tmp = Float64(fma(2.0, y, t) * x);
	else
		tmp = Float64(5.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.2e-31], N[Not[LessEqual[x, 3.4e-52]], $MachinePrecision]], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999991e-31 or 3.40000000000000017e-52 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{5 \cdot y + x \cdot \left(t + 2 \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot y\right) + 5 \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(t + 2 \cdot y\right) \cdot x} + 5 \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t + 2 \cdot y, x, 5 \cdot y\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot y + t}, x, 5 \cdot y\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, y, t\right)}, x, 5 \cdot y\right) \]

      6. lower-*.f64 71.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, \color{blue}{5 \cdot y}\right) \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(t + 2 \cdot y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(2, y, t\right) \cdot \color{blue}{x} \]

   if -5.19999999999999991e-31 < x < 3.40000000000000017e-52

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-31} \lor \neg \left(x \leq 3.4 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+215}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq -2.5:\\ \;\;\;\;\left(2 \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.7e+215)
   (* (* z x) 2.0)
   (if (<= x -2.5) (* (* 2.0 x) y) (if (<= x 3.4e-52) (* 5.0 y) (* t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e+215) {
		tmp = (z * x) * 2.0;
	} else if (x <= -2.5) {
		tmp = (2.0 * x) * y;
	} else if (x <= 3.4e-52) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.7d+215)) then
        tmp = (z * x) * 2.0d0
    else if (x <= (-2.5d0)) then
        tmp = (2.0d0 * x) * y
    else if (x <= 3.4d-52) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.7e+215) {
		tmp = (z * x) * 2.0;
	} else if (x <= -2.5) {
		tmp = (2.0 * x) * y;
	} else if (x <= 3.4e-52) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.7e+215:
		tmp = (z * x) * 2.0
	elif x <= -2.5:
		tmp = (2.0 * x) * y
	elif x <= 3.4e-52:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.7e+215)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (x <= -2.5)
		tmp = Float64(Float64(2.0 * x) * y);
	elseif (x <= 3.4e-52)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.7e+215)
		tmp = (z * x) * 2.0;
	elseif (x <= -2.5)
		tmp = (2.0 * x) * y;
	elseif (x <= 3.4e-52)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.7e+215], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, -2.5], N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 3.4e-52], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.70000000000000009e215

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot z\right) \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

      4. lower-*.f64 56.5

         \[\leadsto \color{blue}{\left(z \cdot x\right)} \cdot 2 \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]

   if -1.70000000000000009e215 < x < -2.5

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      9. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \cdot y \]

      10. associate--r- N/A

          \[\leadsto \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \cdot y \]

      11. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(\color{blue}{2} \cdot x + 5\right) \cdot y \]

      14. lower-fma.f64 45.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(2 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 45.7%

      \[\leadsto \left(2 \cdot x\right) \cdot y \]

   if -2.5 < x < 3.40000000000000017e-52

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.4

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if 3.40000000000000017e-52 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 47.3

         \[\leadsto \color{blue}{t \cdot x} \]

   5. Applied rewrites 47.3%

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

Alternative 9: 47.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+229}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -2.5:\\ \;\;\;\;\left(2 \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-52}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.3e+229)
   (* t x)
   (if (<= x -2.5) (* (* 2.0 x) y) (if (<= x 3.4e-52) (* 5.0 y) (* t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.3e+229) {
		tmp = t * x;
	} else if (x <= -2.5) {
		tmp = (2.0 * x) * y;
	} else if (x <= 3.4e-52) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-3.3d+229)) then
        tmp = t * x
    else if (x <= (-2.5d0)) then
        tmp = (2.0d0 * x) * y
    else if (x <= 3.4d-52) then
        tmp = 5.0d0 * y
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.3e+229) {
		tmp = t * x;
	} else if (x <= -2.5) {
		tmp = (2.0 * x) * y;
	} else if (x <= 3.4e-52) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.3e+229:
		tmp = t * x
	elif x <= -2.5:
		tmp = (2.0 * x) * y
	elif x <= 3.4e-52:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.3e+229)
		tmp = Float64(t * x);
	elseif (x <= -2.5)
		tmp = Float64(Float64(2.0 * x) * y);
	elseif (x <= 3.4e-52)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.3e+229)
		tmp = t * x;
	elseif (x <= -2.5)
		tmp = (2.0 * x) * y;
	elseif (x <= 3.4e-52)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.3e+229], N[(t * x), $MachinePrecision], If[LessEqual[x, -2.5], N[(N[(2.0 * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 3.4e-52], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2999999999999999e229 or 3.40000000000000017e-52 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 46.7

         \[\leadsto \color{blue}{t \cdot x} \]

   5. Applied rewrites 46.7%

      \[\leadsto \color{blue}{t \cdot x} \]

   if -3.2999999999999999e229 < x < -2.5

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right) \cdot y} \]

      9. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \cdot y \]

      10. associate--r- N/A

          \[\leadsto \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \cdot y \]

      11. neg-sub0 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \cdot y \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \left(\color{blue}{2} \cdot x + 5\right) \cdot y \]

      14. lower-fma.f64 45.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right)} \cdot y \]

   5. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 5\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \left(2 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 45.1%

      \[\leadsto \left(2 \cdot x\right) \cdot y \]

   if -2.5 < x < 3.40000000000000017e-52

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.4

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-31} \lor \neg \left(x \leq 3.4 \cdot 10^{-52}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.2e-31) (not (<= x 3.4e-52))) (* t x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e-31) || !(x <= 3.4e-52)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-5.2d-31)) .or. (.not. (x <= 3.4d-52))) then
        tmp = t * x
    else
        tmp = 5.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e-31) || !(x <= 3.4e-52)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.2e-31) or not (x <= 3.4e-52):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.2e-31) || !(x <= 3.4e-52))
		tmp = Float64(t * x);
	else
		tmp = Float64(5.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.2e-31) || ~((x <= 3.4e-52)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.2e-31], N[Not[LessEqual[x, 3.4e-52]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999991e-31 or 3.40000000000000017e-52 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 39.4

         \[\leadsto \color{blue}{t \cdot x} \]

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{t \cdot x} \]

   if -5.19999999999999991e-31 < x < 3.40000000000000017e-52

   1. Initial program 99.7%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 59.6

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-31} \lor \neg \left(x \leq 3.4 \cdot 10^{-52}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ 5 \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 5.0 y))

C

double code(double x, double y, double z, double t) {
	return 5.0 * 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 = 5.0d0 * y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 5.0 * y;
}

Python

def code(x, y, z, t):
	return 5.0 * y

Julia

function code(x, y, z, t)
	return Float64(5.0 * y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 5.0 * y;
end

Wolfram

code[x_, y_, z_, t_] := N[(5.0 * y), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 27.1

      \[\leadsto \color{blue}{5 \cdot y} \]

5. Applied rewrites 27.1%

   \[\leadsto \color{blue}{5 \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))