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

Percentage Accurate: 99.9% → 99.8%

Time: 7.1s

Alternatives: 11

Speedup: 1.0×

• 27.1% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-y}{-1}, \frac{5 \cdot t}{t}, x \cdot \mathsf{fma}\left(z + y, 2, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ (- y) -1.0) (/ (* 5.0 t) t) (* x (fma (+ z y) 2.0 t))))

C

double code(double x, double y, double z, double t) {
	return fma((-y / -1.0), ((5.0 * t) / t), (x * fma((z + y), 2.0, t)));
}

Julia

function code(x, y, z, t)
	return fma(Float64(Float64(-y) / -1.0), Float64(Float64(5.0 * t) / t), Float64(x * fma(Float64(z + y), 2.0, t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-y) / -1.0), $MachinePrecision] * N[(N[(5.0 * t), $MachinePrecision] / t), $MachinePrecision] + N[(x * N[(N[(z + y), $MachinePrecision] * 2.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. distribute-lft-in N/A

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

   2. +-commutative 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. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. cancel-sign-sub N/A

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

5. Applied rewrites 88.8%

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

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\frac{-y}{-1}, \color{blue}{\frac{5 \cdot t}{t}}, x \cdot \mathsf{fma}\left(z + y, 2, t\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -102000000000 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -102000000000.0) (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 <= -102000000000.0) || !(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 <= -102000000000.0) || !(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, -102000000000.0], 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 < -1.02e11 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. distribute-lft-in N/A

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

      2. +-commutative 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. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. cancel-sign-sub N/A

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

   5. Applied rewrites 92.2%

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

   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. distribute-lft-in N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      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 99.5

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

   8. Applied rewrites 99.5%

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

   if -1.02e11 < x < 2.5

   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. 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 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -102000000000 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-60} \lor \neg \left(x \leq 1.75 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.8e-60) (not (<= x 1.75e-65)))
   (* (fma 2.0 (+ z y) t) x)
   (fma y 5.0 (* (+ z z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.8e-60) || !(x <= 1.75e-65)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(y, 5.0, ((z + z) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.8e-60) || !(x <= 1.75e-65))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(y, 5.0, Float64(Float64(z + z) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.8e-60], N[Not[LessEqual[x, 1.75e-65]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(N[(z + z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999994e-60 or 1.75000000000000002e-65 < 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. distribute-lft-in N/A

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

      2. +-commutative 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. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. cancel-sign-sub N/A

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

   5. Applied rewrites 92.2%

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

   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. distribute-lft-in N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      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 97.2

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

   8. Applied rewrites 97.2%

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

   if -3.79999999999999994e-60 < x < 1.75000000000000002e-65

   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. 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 89.2

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

   7. Applied rewrites 89.2%

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

   9. Applied rewrites 89.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-60} \lor \neg \left(x \leq 1.75 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(z + z\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-36} \lor \neg \left(x \leq 1.7 \cdot 10^{-79}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.5e-36) (not (<= x 1.7e-79)))
   (* (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 <= -5.5e-36) || !(x <= 1.7e-79)) {
		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 <= -5.5e-36) || !(x <= 1.7e-79))
		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, -5.5e-36], N[Not[LessEqual[x, 1.7e-79]], $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 < -5.49999999999999984e-36 or 1.69999999999999988e-79 < 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. distribute-lft-in N/A

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

      2. +-commutative 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. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. mul-1-neg N/A

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

      14. cancel-sign-sub N/A

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

   5. Applied rewrites 92.1%

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

   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. distribute-lft-in N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      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 97.8

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

   8. Applied rewrites 97.8%

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

   if -5.49999999999999984e-36 < x < 1.69999999999999988e-79

   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. 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 87.1

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

   7. Applied rewrites 87.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-36} \lor \neg \left(x \leq 1.7 \cdot 10^{-79}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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]

Derivation

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. Add Preprocessing

Alternative 6: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-36} \lor \neg \left(x \leq 9 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.5e-36) (not (<= x 9e-66)))
   (* (fma 2.0 z t) x)
   (fma y 5.0 (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.5e-36) || !(x <= 9e-66)) {
		tmp = fma(2.0, z, t) * x;
	} else {
		tmp = fma(y, 5.0, (x * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.5e-36) || !(x <= 9e-66))
		tmp = Float64(fma(2.0, z, 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, -5.5e-36], N[Not[LessEqual[x, 9e-66]], $MachinePrecision]], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999984e-36 or 8.9999999999999995e-66 < 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 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 75.6

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

   5. Applied rewrites 75.6%

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

   if -5.49999999999999984e-36 < x < 8.9999999999999995e-66

   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. 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 86.3

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

   7. Applied rewrites 86.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-36} \lor \neg \left(x \leq 9 \cdot 10^{-66}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+55} \lor \neg \left(y \leq 1.9 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.12e+55) (not (<= y 1.9e+136)))
   (* (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 <= -1.12e+55) || !(y <= 1.9e+136)) {
		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 <= -1.12e+55) || !(y <= 1.9e+136))
		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, -1.12e+55], N[Not[LessEqual[y, 1.9e+136]], $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 < -1.12000000000000006e55 or 1.90000000000000007e136 < 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 86.5

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

   5. Applied rewrites 86.5%

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

   if -1.12000000000000006e55 < y < 1.90000000000000007e136

   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 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 73.5

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

   5. Applied rewrites 73.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+55} \lor \neg \left(y \leq 1.9 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+166} \lor \neg \left(t \leq 9.3 \cdot 10^{+182}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.1e+166) (not (<= t 9.3e+182)))
   (* t x)
   (* (fma 2.0 x 5.0) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.1e+166) || !(t <= 9.3e+182)) {
		tmp = t * x;
	} else {
		tmp = fma(2.0, x, 5.0) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.1e+166) || !(t <= 9.3e+182))
		tmp = Float64(t * x);
	else
		tmp = Float64(fma(2.0, x, 5.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.1e+166], N[Not[LessEqual[t, 9.3e+182]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.09999999999999983e166 or 9.3000000000000005e182 < t

   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 78.4

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

   5. Applied rewrites 78.4%

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

   if -3.09999999999999983e166 < t < 9.3000000000000005e182

   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 58.3

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

   5. Applied rewrites 58.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+166} \lor \neg \left(t \leq 9.3 \cdot 10^{+182}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+103}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-36}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-63}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.8e+103)
   (* t x)
   (if (<= x -5.4e-36) (* (* z x) 2.0) (if (<= x 5e-63) (* 5.0 y) (* t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.8e+103) {
		tmp = t * x;
	} else if (x <= -5.4e-36) {
		tmp = (z * x) * 2.0;
	} else if (x <= 5e-63) {
		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 <= (-5.8d+103)) then
        tmp = t * x
    else if (x <= (-5.4d-36)) then
        tmp = (z * x) * 2.0d0
    else if (x <= 5d-63) 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 <= -5.8e+103) {
		tmp = t * x;
	} else if (x <= -5.4e-36) {
		tmp = (z * x) * 2.0;
	} else if (x <= 5e-63) {
		tmp = 5.0 * y;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.8e+103:
		tmp = t * x
	elif x <= -5.4e-36:
		tmp = (z * x) * 2.0
	elif x <= 5e-63:
		tmp = 5.0 * y
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.8e+103)
		tmp = Float64(t * x);
	elseif (x <= -5.4e-36)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (x <= 5e-63)
		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 <= -5.8e+103)
		tmp = t * x;
	elseif (x <= -5.4e-36)
		tmp = (z * x) * 2.0;
	elseif (x <= 5e-63)
		tmp = 5.0 * y;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.8e+103], N[(t * x), $MachinePrecision], If[LessEqual[x, -5.4e-36], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, 5e-63], N[(5.0 * y), $MachinePrecision], N[(t * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999997e103 or 5.0000000000000002e-63 < 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 44.7

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

   5. Applied rewrites 44.7%

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

   if -5.7999999999999997e103 < x < -5.40000000000000015e-36

   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 58.9

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

   5. Applied rewrites 58.9%

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

   if -5.40000000000000015e-36 < x < 5.0000000000000002e-63

   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 74.0

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

   5. Applied rewrites 74.0%

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

Alternative 10: 46.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-31} \lor \neg \left(x \leq 5 \cdot 10^{-63}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.85e-31) (not (<= x 5e-63))) (* t x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.85e-31) || !(x <= 5e-63)) {
		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 <= (-1.85d-31)) .or. (.not. (x <= 5d-63))) 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 <= -1.85e-31) || !(x <= 5e-63)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.85e-31) or not (x <= 5e-63):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.85e-31) || !(x <= 5e-63))
		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 <= -1.85e-31) || ~((x <= 5e-63)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.85e-31], N[Not[LessEqual[x, 5e-63]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8499999999999999e-31 or 5.0000000000000002e-63 < 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 38.6

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

   5. Applied rewrites 38.6%

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

   if -1.8499999999999999e-31 < x < 5.0000000000000002e-63

   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 72.5

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

   5. Applied rewrites 72.5%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-31} \lor \neg \left(x \leq 5 \cdot 10^{-63}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.0% 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 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 x around 0

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

   1. lower-*.f64 30.2

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

5. Applied rewrites 30.2%

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

Reproduce

herbie shell --seed 2024324 
(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)))