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

Percentage Accurate: 99.9% → 99.0%

Time: 5.9s

Alternatives: 9

Speedup: N/A×

• 21.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+l+ N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-out N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x, x \cdot z\right)\\ \mathbf{if}\;x \leq -110000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;y \cdot x + z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y x (* x z))))
   (if (<= x -110000.0) t_0 (if (<= x 5.0) (+ (* y x) (* z 5.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, x, (x * z));
	double tmp;
	if (x <= -110000.0) {
		tmp = t_0;
	} else if (x <= 5.0) {
		tmp = (y * x) + (z * 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, x, Float64(x * z))
	tmp = 0.0
	if (x <= -110000.0)
		tmp = t_0;
	elseif (x <= 5.0)
		tmp = Float64(Float64(y * x) + Float64(z * 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * x + N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -110000.0], t$95$0, If[LessEqual[x, 5.0], N[(N[(y * x), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e5 or 5 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if -1.1e5 < x < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 99.2

         \[\leadsto \color{blue}{x \cdot y} + z \cdot 5 \]

   5. Applied rewrites 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -110000:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot z\right)\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;y \cdot x + z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-13}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+201}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.35e-13)
   (* y x)
   (if (<= x 5.0) (* z 5.0) (if (<= x 3.5e+201) (* x z) (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.35e-13) {
		tmp = y * x;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else if (x <= 3.5e+201) {
		tmp = x * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.35d-13)) then
        tmp = y * x
    else if (x <= 5.0d0) then
        tmp = z * 5.0d0
    else if (x <= 3.5d+201) then
        tmp = x * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.35e-13) {
		tmp = y * x;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else if (x <= 3.5e+201) {
		tmp = x * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.35e-13:
		tmp = y * x
	elif x <= 5.0:
		tmp = z * 5.0
	elif x <= 3.5e+201:
		tmp = x * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.35e-13)
		tmp = Float64(y * x);
	elseif (x <= 5.0)
		tmp = Float64(z * 5.0);
	elseif (x <= 3.5e+201)
		tmp = Float64(x * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.35e-13)
		tmp = y * x;
	elseif (x <= 5.0)
		tmp = z * 5.0;
	elseif (x <= 3.5e+201)
		tmp = x * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.35e-13], N[(y * x), $MachinePrecision], If[LessEqual[x, 5.0], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 3.5e+201], N[(x * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3500000000000001e-13 or 3.5000000000000002e201 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if -2.3500000000000001e-13 < x < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if 5 < x < 3.5000000000000002e201

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-13}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+201}:\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x, x \cdot z\right)\\ \mathbf{if}\;x \leq -110000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(z, 5, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y x (* x z))))
   (if (<= x -110000.0) t_0 (if (<= x 5.0) (fma z 5.0 (* y x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, x, (x * z));
	double tmp;
	if (x <= -110000.0) {
		tmp = t_0;
	} else if (x <= 5.0) {
		tmp = fma(z, 5.0, (y * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, x, Float64(x * z))
	tmp = 0.0
	if (x <= -110000.0)
		tmp = t_0;
	elseif (x <= 5.0)
		tmp = fma(z, 5.0, Float64(y * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * x + N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -110000.0], t$95$0, If[LessEqual[x, 5.0], N[(z * 5.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e5 or 5 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if -1.1e5 < x < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 99.2

         \[\leadsto \color{blue}{x \cdot y} + z \cdot 5 \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{x \cdot y} + z \cdot 5 \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot y + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + x \cdot y} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + x \cdot y \]

      4. lower-fma.f64 99.2

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

   7. Applied rewrites 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -110000:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot z\right)\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(z, 5, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x, x \cdot z\right)\\ \mathbf{if}\;x \leq -9.7 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 32000000000000:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y x (* x z))))
   (if (<= x -9.7e-11) t_0 (if (<= x 32000000000000.0) (* z (+ x 5.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, x, (x * z));
	double tmp;
	if (x <= -9.7e-11) {
		tmp = t_0;
	} else if (x <= 32000000000000.0) {
		tmp = z * (x + 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, x, Float64(x * z))
	tmp = 0.0
	if (x <= -9.7e-11)
		tmp = t_0;
	elseif (x <= 32000000000000.0)
		tmp = Float64(z * Float64(x + 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * x + N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.7e-11], t$95$0, If[LessEqual[x, 32000000000000.0], N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7000000000000001e-11 or 3.2e13 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if -9.7000000000000001e-11 < x < 3.2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 75.0

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

   5. Applied rewrites 75.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.7 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot z\right)\\ \mathbf{elif}\;x \leq 32000000000000:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -9.7 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 32000000000000:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))))
   (if (<= x -9.7e-11) t_0 (if (<= x 32000000000000.0) (* z (+ x 5.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -9.7e-11) {
		tmp = t_0;
	} else if (x <= 32000000000000.0) {
		tmp = z * (x + 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + z)
    if (x <= (-9.7d-11)) then
        tmp = t_0
    else if (x <= 32000000000000.0d0) then
        tmp = z * (x + 5.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -9.7e-11) {
		tmp = t_0;
	} else if (x <= 32000000000000.0) {
		tmp = z * (x + 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y + z)
	tmp = 0
	if x <= -9.7e-11:
		tmp = t_0
	elif x <= 32000000000000.0:
		tmp = z * (x + 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + z))
	tmp = 0.0
	if (x <= -9.7e-11)
		tmp = t_0;
	elseif (x <= 32000000000000.0)
		tmp = Float64(z * Float64(x + 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y + z);
	tmp = 0.0;
	if (x <= -9.7e-11)
		tmp = t_0;
	elseif (x <= 32000000000000.0)
		tmp = z * (x + 5.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.7e-11], t$95$0, If[LessEqual[x, 32000000000000.0], N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7000000000000001e-11 or 3.2e13 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -9.7000000000000001e-11 < x < 3.2e13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 75.0

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

   5. Applied rewrites 75.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.7 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 32000000000000:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))))
   (if (<= x -2.35e-13) t_0 (if (<= x 6.8e-9) (* z 5.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -2.35e-13) {
		tmp = t_0;
	} else if (x <= 6.8e-9) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + z)
    if (x <= (-2.35d-13)) then
        tmp = t_0
    else if (x <= 6.8d-9) then
        tmp = z * 5.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -2.35e-13) {
		tmp = t_0;
	} else if (x <= 6.8e-9) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y + z)
	tmp = 0
	if x <= -2.35e-13:
		tmp = t_0
	elif x <= 6.8e-9:
		tmp = z * 5.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + z))
	tmp = 0.0
	if (x <= -2.35e-13)
		tmp = t_0;
	elseif (x <= 6.8e-9)
		tmp = Float64(z * 5.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y + z);
	tmp = 0.0;
	if (x <= -2.35e-13)
		tmp = t_0;
	elseif (x <= 6.8e-9)
		tmp = z * 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.35e-13], t$95$0, If[LessEqual[x, 6.8e-9], N[(z * 5.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e-13 or 6.7999999999999997e-9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -2.3500000000000001e-13 < x < 6.7999999999999997e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -110000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -110000.0) (* x z) (if (<= x 5.0) (* z 5.0) (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -110000.0) {
		tmp = x * z;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-110000.0d0)) then
        tmp = x * z
    else if (x <= 5.0d0) then
        tmp = z * 5.0d0
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -110000.0) {
		tmp = x * z;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -110000.0:
		tmp = x * z
	elif x <= 5.0:
		tmp = z * 5.0
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -110000.0)
		tmp = Float64(x * z);
	elseif (x <= 5.0)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -110000.0)
		tmp = x * z;
	elseif (x <= 5.0)
		tmp = z * 5.0;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -110000.0], N[(x * z), $MachinePrecision], If[LessEqual[x, 5.0], N[(z * 5.0), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e5 or 5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.5%

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

   if -1.1e5 < x < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -110000:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * z), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 26.0%

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

9. Final simplification 26.0%

   \[\leadsto x \cdot z \]
10. Add Preprocessing

Developer Target 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (+ x 5) z) (* x y)))

  (+ (* x (+ y z)) (* z 5.0)))