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

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 10

Speedup: 1.2×

• 26.1% 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
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. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. count-2 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 97.7

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. count-2 N/A

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

   8. associate-+l+ N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-rgt-in N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. associate-+l+ N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. associate-+l+ N/A

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

   18. +-commutative N/A

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

   19. count-2 N/A

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

   20. lower-fma.f64 N/A

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

   21. lower-+.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 58.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot z\right) \cdot 2\\ t_2 := \mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{if}\;z \leq -5.7 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{+127}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+161}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* x z) 2.0)) (t_2 (* (fma 2.0 y t) x)))
   (if (<= z -5.7e+179)
     t_1
     (if (<= z -7.6e+127)
       t_2
       (if (<= z -1.22e-207)
         (* (fma 2.0 x 5.0) y)
         (if (<= z 8.8e+161) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * z) * 2.0;
	double t_2 = fma(2.0, y, t) * x;
	double tmp;
	if (z <= -5.7e+179) {
		tmp = t_1;
	} else if (z <= -7.6e+127) {
		tmp = t_2;
	} else if (z <= -1.22e-207) {
		tmp = fma(2.0, x, 5.0) * y;
	} else if (z <= 8.8e+161) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * z) * 2.0)
	t_2 = Float64(fma(2.0, y, t) * x)
	tmp = 0.0
	if (z <= -5.7e+179)
		tmp = t_1;
	elseif (z <= -7.6e+127)
		tmp = t_2;
	elseif (z <= -1.22e-207)
		tmp = Float64(fma(2.0, x, 5.0) * y);
	elseif (z <= 8.8e+161)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -5.7e+179], t$95$1, If[LessEqual[z, -7.6e+127], t$95$2, If[LessEqual[z, -1.22e-207], N[(N[(2.0 * x + 5.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 8.8e+161], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.7e179 or 8.7999999999999999e161 < z

   1. Initial program 98.4%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 84.7

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

   5. Applied rewrites 84.7%

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

   if -5.7e179 < z < -7.59999999999999961e127 or -1.22e-207 < z < 8.7999999999999999e161

   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 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.3%

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

   if -7.59999999999999961e127 < z < -1.22e-207

   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 62.9

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

   5. Applied rewrites 62.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{+179}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 2\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-207}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 5, \mathsf{fma}\left(2, z, t\right) \cdot x\right)\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y 5.0 (* (fma 2.0 z t) x))))
   (if (<= z -1.95e+90)
     t_1
     (if (<= z 2.5e-130) (fma (fma 2.0 y t) x (* 5.0 y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, 5.0, (fma(2.0, z, t) * x));
	double tmp;
	if (z <= -1.95e+90) {
		tmp = t_1;
	} else if (z <= 2.5e-130) {
		tmp = fma(fma(2.0, y, t), x, (5.0 * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, 5.0, Float64(fma(2.0, z, t) * x))
	tmp = 0.0
	if (z <= -1.95e+90)
		tmp = t_1;
	elseif (z <= 2.5e-130)
		tmp = fma(fma(2.0, y, t), x, Float64(5.0 * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * 5.0 + N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.95e+90], t$95$1, If[LessEqual[z, 2.5e-130], N[(N[(2.0 * y + t), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9500000000000001e90 or 2.4999999999999998e-130 < z

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.2

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

      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 95.6

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

   4. Applied rewrites 95.6%

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

   if -1.9500000000000001e90 < z < 2.4999999999999998e-130

   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 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 4: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+206}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2, y, t\right), x, 5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 z t) x)))
   (if (<= z -1.4e+206)
     t_1
     (if (<= z 1.7e+98) (fma (fma 2.0 y t) x (* 5.0 y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, z, t) * x;
	double tmp;
	if (z <= -1.4e+206) {
		tmp = t_1;
	} else if (z <= 1.7e+98) {
		tmp = fma(fma(2.0, y, t), x, (5.0 * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(2.0, z, t) * x)
	tmp = 0.0
	if (z <= -1.4e+206)
		tmp = t_1;
	elseif (z <= 1.7e+98)
		tmp = fma(fma(2.0, y, t), x, Float64(5.0 * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -1.4e+206], t$95$1, If[LessEqual[z, 1.7e+98], N[(N[(2.0 * y + t), $MachinePrecision] * x + N[(5.0 * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e206 or 1.69999999999999986e98 < z

   1. Initial program 98.6%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 96.0

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

   5. Applied rewrites 96.0%

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

   if -1.3999999999999999e206 < z < 1.69999999999999986e98

   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 90.1

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

   5. Applied rewrites 90.1%

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

Alternative 5: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot z\right) \cdot 2\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-203}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+117}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* x z) 2.0)))
   (if (<= z -2.2e+90)
     t_1
     (if (<= z -7.4e-203) (* 5.0 y) (if (<= z 7.6e+117) (* x t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * z) * 2.0;
	double tmp;
	if (z <= -2.2e+90) {
		tmp = t_1;
	} else if (z <= -7.4e-203) {
		tmp = 5.0 * y;
	} else if (z <= 7.6e+117) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * z) * 2.0d0
    if (z <= (-2.2d+90)) then
        tmp = t_1
    else if (z <= (-7.4d-203)) then
        tmp = 5.0d0 * y
    else if (z <= 7.6d+117) then
        tmp = x * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * z) * 2.0;
	double tmp;
	if (z <= -2.2e+90) {
		tmp = t_1;
	} else if (z <= -7.4e-203) {
		tmp = 5.0 * y;
	} else if (z <= 7.6e+117) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * z) * 2.0
	tmp = 0
	if z <= -2.2e+90:
		tmp = t_1
	elif z <= -7.4e-203:
		tmp = 5.0 * y
	elif z <= 7.6e+117:
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * z) * 2.0)
	tmp = 0.0
	if (z <= -2.2e+90)
		tmp = t_1;
	elseif (z <= -7.4e-203)
		tmp = Float64(5.0 * y);
	elseif (z <= 7.6e+117)
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * z) * 2.0;
	tmp = 0.0;
	if (z <= -2.2e+90)
		tmp = t_1;
	elseif (z <= -7.4e-203)
		tmp = 5.0 * y;
	elseif (z <= 7.6e+117)
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[z, -2.2e+90], t$95$1, If[LessEqual[z, -7.4e-203], N[(5.0 * y), $MachinePrecision], If[LessEqual[z, 7.6e+117], N[(x * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1999999999999999e90 or 7.6000000000000003e117 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 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 71.9

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

   5. Applied rewrites 71.9%

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

   if -2.1999999999999999e90 < z < -7.40000000000000004e-203

   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 43.3

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

   5. Applied rewrites 43.3%

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

   if -7.40000000000000004e-203 < z < 7.6000000000000003e117

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 44.0

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

   5. Applied rewrites 44.0%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+90}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 2\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-203}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+117}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 5, \left(2 \cdot y\right) \cdot x\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y 5.0 (* (* 2.0 y) x))))
   (if (<= y -5.5e-11) t_1 (if (<= y 1.2e+74) (* (fma 2.0 z t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, 5.0, ((2.0 * y) * x));
	double tmp;
	if (y <= -5.5e-11) {
		tmp = t_1;
	} else if (y <= 1.2e+74) {
		tmp = fma(2.0, z, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, 5.0, Float64(Float64(2.0 * y) * x))
	tmp = 0.0
	if (y <= -5.5e-11)
		tmp = t_1;
	elseif (y <= 1.2e+74)
		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[(y * 5.0 + N[(N[(2.0 * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e-11], t$95$1, If[LessEqual[y, 1.2e+74], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999975e-11 or 1.20000000000000004e74 < 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. 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. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 96.9

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. count-2 N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. associate-+l+ N/A

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

      18. +-commutative N/A

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

      19. count-2 N/A

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

      20. lower-fma.f64 N/A

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

      21. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 83.9

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

   9. Applied rewrites 83.9%

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

   if -5.49999999999999975e-11 < y < 1.20000000000000004e74

   1. Initial program 99.3%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 7: 77.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, x, 5\right) \cdot y\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(2, z, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (fma 2.0 x 5.0) y)))
   (if (<= y -5.5e-11) t_1 (if (<= y 1.2e+74) (* (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 <= -5.5e-11) {
		tmp = t_1;
	} else if (y <= 1.2e+74) {
		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 <= -5.5e-11)
		tmp = t_1;
	elseif (y <= 1.2e+74)
		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, -5.5e-11], t$95$1, If[LessEqual[y, 1.2e+74], N[(N[(2.0 * z + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999975e-11 or 1.20000000000000004e74 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. neg-sub0 N/A

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

      5. associate--r- N/A

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

      6. neg-sub0 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. associate--r- N/A

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

      11. neg-sub0 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -5.49999999999999975e-11 < y < 1.20000000000000004e74

   1. Initial program 99.3%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 8: 57.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot z\right) \cdot 2\\ \mathbf{if}\;z \leq -5.7 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* x z) 2.0)))
   (if (<= z -5.7e+179) t_1 (if (<= z 8.8e+161) (* (fma 2.0 y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * z) * 2.0;
	double tmp;
	if (z <= -5.7e+179) {
		tmp = t_1;
	} else if (z <= 8.8e+161) {
		tmp = fma(2.0, y, t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * z) * 2.0)
	tmp = 0.0
	if (z <= -5.7e+179)
		tmp = t_1;
	elseif (z <= 8.8e+161)
		tmp = Float64(fma(2.0, y, t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[z, -5.7e+179], t$95$1, If[LessEqual[z, 8.8e+161], N[(N[(2.0 * y + t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.7e179 or 8.7999999999999999e161 < z

   1. Initial program 98.4%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 84.7

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

   5. Applied rewrites 84.7%

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

   if -5.7e179 < z < 8.7999999999999999e161

   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 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{+179}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 2\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(2, y, t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+38}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.2e-26) (* 5.0 y) (if (<= y 7e+38) (* x t) (* 5.0 y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.2e-26) {
		tmp = 5.0 * y;
	} else if (y <= 7e+38) {
		tmp = x * t;
	} 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 (y <= (-3.2d-26)) then
        tmp = 5.0d0 * y
    else if (y <= 7d+38) then
        tmp = x * t
    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 (y <= -3.2e-26) {
		tmp = 5.0 * y;
	} else if (y <= 7e+38) {
		tmp = x * t;
	} else {
		tmp = 5.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.2e-26:
		tmp = 5.0 * y
	elif y <= 7e+38:
		tmp = x * t
	else:
		tmp = 5.0 * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.2e-26)
		tmp = Float64(5.0 * y);
	elseif (y <= 7e+38)
		tmp = Float64(x * t);
	else
		tmp = Float64(5.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.2e-26)
		tmp = 5.0 * y;
	elseif (y <= 7e+38)
		tmp = x * t;
	else
		tmp = 5.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.2e-26], N[(5.0 * y), $MachinePrecision], If[LessEqual[y, 7e+38], N[(x * t), $MachinePrecision], N[(5.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2000000000000001e-26 or 7.00000000000000003e38 < 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 x around 0

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

      1. lower-*.f64 49.5

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

   5. Applied rewrites 49.5%

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

   if -3.2000000000000001e-26 < y < 7.00000000000000003e38

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+38}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.0% 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 99.6%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 26.4

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

5. Applied rewrites 26.4%

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

Reproduce

herbie shell --seed 2024332 
(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)))