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

Percentage Accurate: 99.9% → 100.0%

Time: 8.2s

Alternatives: 12

Speedup: 1.2×

• 27.3% 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: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(N[(2.0 * N[(z + y), $MachinePrecision] + 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. 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. count-2 N/A

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

   15. lower-fma.f64 100.0

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 (+ z y) t) x)))
   (if (<= x -2.9e-26)
     t_1
     (if (<= x -1e-254)
       (fma 5.0 y (* t x))
       (if (<= x 7e-73) (fma y 5.0 (* (* 2.0 z) x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, (z + y), t) * x;
	double tmp;
	if (x <= -2.9e-26) {
		tmp = t_1;
	} else if (x <= -1e-254) {
		tmp = fma(5.0, y, (t * x));
	} else if (x <= 7e-73) {
		tmp = fma(y, 5.0, ((2.0 * z) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, Float64(z + y), t) * x)
	tmp = 0.0
	if (x <= -2.9e-26)
		tmp = t_1;
	elseif (x <= -1e-254)
		tmp = fma(5.0, y, Float64(t * x));
	elseif (x <= 7e-73)
		tmp = fma(y, 5.0, Float64(Float64(2.0 * z) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.9e-26], t$95$1, If[LessEqual[x, -1e-254], N[(5.0 * y + N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-73], N[(y * 5.0 + N[(N[(2.0 * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8999999999999998e-26 or 6.9999999999999995e-73 < 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 x around 0

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

      1. lower-*.f64 3.5

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

   5. Applied rewrites 3.5%

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

   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 97.9

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

   8. Applied rewrites 97.9%

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

   if -2.8999999999999998e-26 < x < -9.9999999999999991e-255

   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. count-2 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. associate-+l+ N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. count-2-rev N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 82.2

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

   9. Applied rewrites 82.2%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 82.2%

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

   if -9.9999999999999991e-255 < x < 6.9999999999999995e-73

   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. count-2 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \left(2 \cdot z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+34} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.3e+34) (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 <= -2.3e+34) || !(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 <= -2.3e+34) || !(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, -2.3e+34], 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 < -2.2999999999999998e34 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 x around 0

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

      1. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   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 99.2

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

   8. Applied rewrites 99.2%

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

   if -2.2999999999999998e34 < 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. count-2 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+34} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+90} \lor \neg \left(z \leq 14.2\right):\\ \;\;\;\;\mathsf{fma}\left(x + x, z + y, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.4e+90) (not (<= z 14.2)))
   (fma (+ x x) (+ z y) (* 5.0 y))
   (fma (fma 2.0 y t) x (* 5.0 y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e+90) || !(z <= 14.2)) {
		tmp = fma((x + x), (z + y), (5.0 * y));
	} else {
		tmp = fma(fma(2.0, y, t), x, (5.0 * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.4e+90) || !(z <= 14.2))
		tmp = fma(Float64(x + x), Float64(z + y), Float64(5.0 * y));
	else
		tmp = fma(fma(2.0, y, t), x, Float64(5.0 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.4e+90], N[Not[LessEqual[z, 14.2]], $MachinePrecision]], N[(N[(x + x), $MachinePrecision] * N[(z + y), $MachinePrecision] + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * y + t), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000018e90 or 14.199999999999999 < z

   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 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 89.9%

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

   if -3.40000000000000018e90 < z < 14.199999999999999

   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 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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+90} \lor \neg \left(z \leq 14.2\right):\\ \;\;\;\;\mathsf{fma}\left(x + x, z + y, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -8.5e+127)
     t_1
     (if (<= y -1.12e-38)
       (fma 5.0 y (* t x))
       (if (<= y 2.3e-33) (* (fma 2.0 z t) x) 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 <= -8.5e+127) {
		tmp = t_1;
	} else if (y <= -1.12e-38) {
		tmp = fma(5.0, y, (t * x));
	} else if (y <= 2.3e-33) {
		tmp = fma(2.0, z, t) * x;
	} 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 <= -8.5e+127)
		tmp = t_1;
	elseif (y <= -1.12e-38)
		tmp = fma(5.0, y, Float64(t * x));
	elseif (y <= 2.3e-33)
		tmp = Float64(fma(2.0, z, t) * x);
	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, -8.5e+127], t$95$1, If[LessEqual[y, -1.12e-38], N[(5.0 * y + N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-33], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999997e127 or 2.29999999999999986e-33 < 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if -8.4999999999999997e127 < y < -1.1200000000000001e-38

   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. count-2 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. associate-+l+ N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. count-2-rev N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   6. Applied rewrites 97.1%

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

   7. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 85.9

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

   9. Applied rewrites 85.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 77.8%

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

   if -1.1200000000000001e-38 < y < 2.29999999999999986e-33

   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-261}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-34}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -1.95e-57)
     t_1
     (if (<= y -2.1e-261) (* (* z x) 2.0) (if (<= y 4.2e-34) (* t x) 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 <= -1.95e-57) {
		tmp = t_1;
	} else if (y <= -2.1e-261) {
		tmp = (z * x) * 2.0;
	} else if (y <= 4.2e-34) {
		tmp = t * x;
	} 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 <= -1.95e-57)
		tmp = t_1;
	elseif (y <= -2.1e-261)
		tmp = Float64(Float64(z * x) * 2.0);
	elseif (y <= 4.2e-34)
		tmp = Float64(t * x);
	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, -1.95e-57], t$95$1, If[LessEqual[y, -2.1e-261], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 4.2e-34], N[(t * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.95000000000000003e-57 or 4.2000000000000002e-34 < 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.95000000000000003e-57 < y < -2.09999999999999996e-261

   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 54.9

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

   5. Applied rewrites 54.9%

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

   if -2.09999999999999996e-261 < y < 4.2000000000000002e-34

   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 53.3

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

   5. Applied rewrites 53.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-261}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-34}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-22}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-80}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e+33)
   (* (* x y) 2.0)
   (if (<= x -6.2e-22)
     (* t x)
     (if (<= x 1.06e-80) (* 5.0 y) (* (* z x) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+33) {
		tmp = (x * y) * 2.0;
	} else if (x <= -6.2e-22) {
		tmp = t * x;
	} else if (x <= 1.06e-80) {
		tmp = 5.0 * y;
	} else {
		tmp = (z * x) * 2.0;
	}
	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 <= (-1.1d+33)) then
        tmp = (x * y) * 2.0d0
    else if (x <= (-6.2d-22)) then
        tmp = t * x
    else if (x <= 1.06d-80) then
        tmp = 5.0d0 * y
    else
        tmp = (z * x) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+33) {
		tmp = (x * y) * 2.0;
	} else if (x <= -6.2e-22) {
		tmp = t * x;
	} else if (x <= 1.06e-80) {
		tmp = 5.0 * y;
	} else {
		tmp = (z * x) * 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e+33:
		tmp = (x * y) * 2.0
	elif x <= -6.2e-22:
		tmp = t * x
	elif x <= 1.06e-80:
		tmp = 5.0 * y
	else:
		tmp = (z * x) * 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e+33)
		tmp = Float64(Float64(x * y) * 2.0);
	elseif (x <= -6.2e-22)
		tmp = Float64(t * x);
	elseif (x <= 1.06e-80)
		tmp = Float64(5.0 * y);
	else
		tmp = Float64(Float64(z * x) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.1e+33)
		tmp = (x * y) * 2.0;
	elseif (x <= -6.2e-22)
		tmp = t * x;
	elseif (x <= 1.06e-80)
		tmp = 5.0 * y;
	else
		tmp = (z * x) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e+33], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[x, -6.2e-22], N[(t * x), $MachinePrecision], If[LessEqual[x, 1.06e-80], N[(5.0 * y), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.09999999999999997e33

   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 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 49.5%

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

   if -1.09999999999999997e33 < x < -6.20000000000000025e-22

   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 64.4

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

   5. Applied rewrites 64.4%

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

   if -6.20000000000000025e-22 < x < 1.0600000000000001e-80

   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 65.8

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

   5. Applied rewrites 65.8%

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

   if 1.0600000000000001e-80 < 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 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 42.1

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

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-22}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-80}:\\ \;\;\;\;5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-26} \lor \neg \left(x \leq 1.06 \cdot 10^{-80}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.9e-26) (not (<= x 1.06e-80)))
   (* (fma 2.0 (+ z y) t) x)
   (fma 5.0 y (* t x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.9e-26) || !(x <= 1.06e-80)) {
		tmp = fma(2.0, (z + y), t) * x;
	} else {
		tmp = fma(5.0, y, (t * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.9e-26) || !(x <= 1.06e-80))
		tmp = Float64(fma(2.0, Float64(z + y), t) * x);
	else
		tmp = fma(5.0, y, Float64(t * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.9e-26], N[Not[LessEqual[x, 1.06e-80]], $MachinePrecision]], N[(N[(2.0 * N[(z + y), $MachinePrecision] + t), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * y + N[(t * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999998e-26 or 1.0600000000000001e-80 < 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 x around 0

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

      1. lower-*.f64 4.2

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

   5. Applied rewrites 4.2%

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

   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 97.3

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

   8. Applied rewrites 97.3%

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

   if -2.8999999999999998e-26 < x < 1.0600000000000001e-80

   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. count-2 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. associate-+l+ N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. count-2-rev N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 81.6

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

   9. Applied rewrites 81.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 81.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-26} \lor \neg \left(x \leq 1.06 \cdot 10^{-80}\right):\\ \;\;\;\;\mathsf{fma}\left(2, z + y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, y, t \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-22} \lor \neg \left(x \leq 7 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e+33)
   (* (* x y) 2.0)
   (if (or (<= x -6.2e-22) (not (<= x 7e-73))) (* t x) (* 5.0 y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e+33) {
		tmp = (x * y) * 2.0;
	} else if ((x <= -6.2e-22) || !(x <= 7e-73)) {
		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 <= (-1.1d+33)) then
        tmp = (x * y) * 2.0d0
    else if ((x <= (-6.2d-22)) .or. (.not. (x <= 7d-73))) 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 <= -1.1e+33) {
		tmp = (x * y) * 2.0;
	} else if ((x <= -6.2e-22) || !(x <= 7e-73)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e+33:
		tmp = (x * y) * 2.0
	elif (x <= -6.2e-22) or not (x <= 7e-73):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e+33)
		tmp = Float64(Float64(x * y) * 2.0);
	elseif ((x <= -6.2e-22) || !(x <= 7e-73))
		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 <= -1.1e+33)
		tmp = (x * y) * 2.0;
	elseif ((x <= -6.2e-22) || ~((x <= 7e-73)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e+33], N[(N[(x * y), $MachinePrecision] * 2.0), $MachinePrecision], If[Or[LessEqual[x, -6.2e-22], N[Not[LessEqual[x, 7e-73]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999997e33

   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 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 49.5%

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

   if -1.09999999999999997e33 < x < -6.20000000000000025e-22 or 6.9999999999999995e-73 < 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 38.4

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

   5. Applied rewrites 38.4%

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

   if -6.20000000000000025e-22 < x < 6.9999999999999995e-73

   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 65.5

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

   5. Applied rewrites 65.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 2\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-22} \lor \neg \left(x \leq 7 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -57 \lor \neg \left(y \leq 2.3 \cdot 10^{-33}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -57.0) (not (<= y 2.3e-33)))
   (* (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 <= -57.0) || !(y <= 2.3e-33)) {
		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 <= -57.0) || !(y <= 2.3e-33))
		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, -57.0], N[Not[LessEqual[y, 2.3e-33]], $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 < -57 or 2.29999999999999986e-33 < 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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -57 < y < 2.29999999999999986e-33

   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 83.6

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

   5. Applied rewrites 83.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -57 \lor \neg \left(y \leq 2.3 \cdot 10^{-33}\right):\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-22} \lor \neg \left(x \leq 7 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.2e-22) (not (<= x 7e-73))) (* t x) (* 5.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.2e-22) || !(x <= 7e-73)) {
		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 <= (-6.2d-22)) .or. (.not. (x <= 7d-73))) 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 <= -6.2e-22) || !(x <= 7e-73)) {
		tmp = t * x;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.2e-22) or not (x <= 7e-73):
		tmp = t * x
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.2e-22) || !(x <= 7e-73))
		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 <= -6.2e-22) || ~((x <= 7e-73)))
		tmp = t * x;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.2e-22], N[Not[LessEqual[x, 7e-73]], $MachinePrecision]], N[(t * x), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000025e-22 or 6.9999999999999995e-73 < 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 37.4

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

   5. Applied rewrites 37.4%

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

   if -6.20000000000000025e-22 < x < 6.9999999999999995e-73

   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 65.5

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

   5. Applied rewrites 65.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-22} \lor \neg \left(x \leq 7 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.3% 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 32.8

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

5. Applied rewrites 32.8%

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

6. Final simplification 32.8%

   \[\leadsto 5 \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(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)))