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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 11

Speedup: 1.2×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. *-inverses N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 89.8%

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

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 47.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-47}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-20}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+194}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* 2.0 y) x)))
   (if (<= x -3.1e+116)
     t_1
     (if (<= x -1.6e-47)
       (* t x)
       (if (<= x 1.28e-20) (* 5.0 y) (if (<= x 3e+194) (* t x) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 * y) * x;
	double tmp;
	if (x <= -3.1e+116) {
		tmp = t_1;
	} else if (x <= -1.6e-47) {
		tmp = t * x;
	} else if (x <= 1.28e-20) {
		tmp = 5.0 * y;
	} else if (x <= 3e+194) {
		tmp = t * x;
	} 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 = (2.0d0 * y) * x
    if (x <= (-3.1d+116)) then
        tmp = t_1
    else if (x <= (-1.6d-47)) then
        tmp = t * x
    else if (x <= 1.28d-20) then
        tmp = 5.0d0 * y
    else if (x <= 3d+194) then
        tmp = t * x
    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 = (2.0 * y) * x;
	double tmp;
	if (x <= -3.1e+116) {
		tmp = t_1;
	} else if (x <= -1.6e-47) {
		tmp = t * x;
	} else if (x <= 1.28e-20) {
		tmp = 5.0 * y;
	} else if (x <= 3e+194) {
		tmp = t * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 * y) * x
	tmp = 0
	if x <= -3.1e+116:
		tmp = t_1
	elif x <= -1.6e-47:
		tmp = t * x
	elif x <= 1.28e-20:
		tmp = 5.0 * y
	elif x <= 3e+194:
		tmp = t * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 * y) * x)
	tmp = 0.0
	if (x <= -3.1e+116)
		tmp = t_1;
	elseif (x <= -1.6e-47)
		tmp = Float64(t * x);
	elseif (x <= 1.28e-20)
		tmp = Float64(5.0 * y);
	elseif (x <= 3e+194)
		tmp = Float64(t * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 * y) * x;
	tmp = 0.0;
	if (x <= -3.1e+116)
		tmp = t_1;
	elseif (x <= -1.6e-47)
		tmp = t * x;
	elseif (x <= 1.28e-20)
		tmp = 5.0 * y;
	elseif (x <= 3e+194)
		tmp = t * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.1e+116], t$95$1, If[LessEqual[x, -1.6e-47], N[(t * x), $MachinePrecision], If[LessEqual[x, 1.28e-20], N[(5.0 * y), $MachinePrecision], If[LessEqual[x, 3e+194], N[(t * x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.09999999999999996e116 or 3.0000000000000003e194 < x

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 39.3%

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

   if -3.09999999999999996e116 < x < -1.6e-47 or 1.2800000000000001e-20 < x < 3.0000000000000003e194

   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 34.5

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

   5. Applied rewrites 34.5%

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

   if -1.6e-47 < x < 1.2800000000000001e-20

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+116}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-47}:\\ \;\;\;\;t \cdot x\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{-20}:\\ \;\;\;\;5 \cdot y\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+194}:\\ \;\;\;\;t \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))