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

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 7

Speedup: 1.1×

• 21.2% 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: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;x \leq 11600000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.8e-79)
   (* x y)
   (if (<= x 1.25e-29)
     (* 5.0 z)
     (if (<= x 11600000000000.0) (* x y) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.8e-79) {
		tmp = x * y;
	} else if (x <= 1.25e-29) {
		tmp = 5.0 * z;
	} else if (x <= 11600000000000.0) {
		tmp = x * y;
	} 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 <= (-9.8d-79)) then
        tmp = x * y
    else if (x <= 1.25d-29) then
        tmp = 5.0d0 * z
    else if (x <= 11600000000000.0d0) then
        tmp = x * y
    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 <= -9.8e-79) {
		tmp = x * y;
	} else if (x <= 1.25e-29) {
		tmp = 5.0 * z;
	} else if (x <= 11600000000000.0) {
		tmp = x * y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.8e-79:
		tmp = x * y
	elif x <= 1.25e-29:
		tmp = 5.0 * z
	elif x <= 11600000000000.0:
		tmp = x * y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.8e-79)
		tmp = Float64(x * y);
	elseif (x <= 1.25e-29)
		tmp = Float64(5.0 * z);
	elseif (x <= 11600000000000.0)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.8e-79)
		tmp = x * y;
	elseif (x <= 1.25e-29)
		tmp = 5.0 * z;
	elseif (x <= 11600000000000.0)
		tmp = x * y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.8e-79], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.25e-29], N[(5.0 * z), $MachinePrecision], If[LessEqual[x, 11600000000000.0], N[(x * y), $MachinePrecision], N[(x * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000001e-79 or 1.24999999999999996e-29 < x < 1.16e13

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if -9.8000000000000001e-79 < x < 1.24999999999999996e-29

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 1.16e13 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;x \leq 11600000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -560:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))))
   (if (<= x -560.0) t_0 (if (<= x 9.5e-6) (fma z 5.0 (* x y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -560.0) {
		tmp = t_0;
	} else if (x <= 9.5e-6) {
		tmp = fma(z, 5.0, (x * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + z))
	tmp = 0.0
	if (x <= -560.0)
		tmp = t_0;
	elseif (x <= 9.5e-6)
		tmp = fma(z, 5.0, Float64(x * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -560.0], t$95$0, If[LessEqual[x, 9.5e-6], N[(z * 5.0 + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -560 or 9.5000000000000005e-6 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -560 < x < 9.5000000000000005e-6

   1. Initial program 99.8%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))))
   (if (<= x -5.5e-71) t_0 (if (<= x 3.4e-31) (* 5.0 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -5.5e-71) {
		tmp = t_0;
	} else if (x <= 3.4e-31) {
		tmp = 5.0 * z;
	} 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 <= (-5.5d-71)) then
        tmp = t_0
    else if (x <= 3.4d-31) then
        tmp = 5.0d0 * z
    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 <= -5.5e-71) {
		tmp = t_0;
	} else if (x <= 3.4e-31) {
		tmp = 5.0 * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y + z)
	tmp = 0
	if x <= -5.5e-71:
		tmp = t_0
	elif x <= 3.4e-31:
		tmp = 5.0 * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + z))
	tmp = 0.0
	if (x <= -5.5e-71)
		tmp = t_0;
	elseif (x <= 3.4e-31)
		tmp = Float64(5.0 * z);
	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 <= -5.5e-71)
		tmp = t_0;
	elseif (x <= 3.4e-31)
		tmp = 5.0 * z;
	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, -5.5e-71], t$95$0, If[LessEqual[x, 3.4e-31], N[(5.0 * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999997e-71 or 3.4000000000000001e-31 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -5.4999999999999997e-71 < x < 3.4000000000000001e-31

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -560.0) (* x z) (if (<= x 9.5e-6) (* 5.0 z) (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -560.0) {
		tmp = x * z;
	} else if (x <= 9.5e-6) {
		tmp = 5.0 * z;
	} 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 <= (-560.0d0)) then
        tmp = x * z
    else if (x <= 9.5d-6) then
        tmp = 5.0d0 * z
    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 <= -560.0) {
		tmp = x * z;
	} else if (x <= 9.5e-6) {
		tmp = 5.0 * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -560.0:
		tmp = x * z
	elif x <= 9.5e-6:
		tmp = 5.0 * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -560.0)
		tmp = Float64(x * z);
	elseif (x <= 9.5e-6)
		tmp = Float64(5.0 * z);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -560.0)
		tmp = x * z;
	elseif (x <= 9.5e-6)
		tmp = 5.0 * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -560.0], N[(x * z), $MachinePrecision], If[LessEqual[x, 9.5e-6], N[(5.0 * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -560 or 9.5000000000000005e-6 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.2%

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

   if -560 < x < 9.5000000000000005e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -560:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
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 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 7: 27.1% 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 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 64.0

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

5. Applied rewrites 64.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 28.7%

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

9. Final simplification 28.7%

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

Developer Target 1: 97.6% 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 2024277 
(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)))