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

Percentage Accurate: 99.9% → 100.0%

Time: 6.5s

Alternatives: 8

Speedup: 1.1×

• 21.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 61.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+18}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+178}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+18)
   (* z x)
   (if (<= x -1.75e-51)
     (* x y)
     (if (<= x 5.0) (* z 5.0) (if (<= x 3.2e+178) (* z x) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+18) {
		tmp = z * x;
	} else if (x <= -1.75e-51) {
		tmp = x * y;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else if (x <= 3.2e+178) {
		tmp = z * x;
	} else {
		tmp = x * y;
	}
	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 <= (-9d+18)) then
        tmp = z * x
    else if (x <= (-1.75d-51)) then
        tmp = x * y
    else if (x <= 5.0d0) then
        tmp = z * 5.0d0
    else if (x <= 3.2d+178) then
        tmp = z * x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+18) {
		tmp = z * x;
	} else if (x <= -1.75e-51) {
		tmp = x * y;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else if (x <= 3.2e+178) {
		tmp = z * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9e+18:
		tmp = z * x
	elif x <= -1.75e-51:
		tmp = x * y
	elif x <= 5.0:
		tmp = z * 5.0
	elif x <= 3.2e+178:
		tmp = z * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+18)
		tmp = Float64(z * x);
	elseif (x <= -1.75e-51)
		tmp = Float64(x * y);
	elseif (x <= 5.0)
		tmp = Float64(z * 5.0);
	elseif (x <= 3.2e+178)
		tmp = Float64(z * x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9e+18)
		tmp = z * x;
	elseif (x <= -1.75e-51)
		tmp = x * y;
	elseif (x <= 5.0)
		tmp = z * 5.0;
	elseif (x <= 3.2e+178)
		tmp = z * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+18], N[(z * x), $MachinePrecision], If[LessEqual[x, -1.75e-51], N[(x * y), $MachinePrecision], If[LessEqual[x, 5.0], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 3.2e+178], N[(z * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e18 or 5 < x < 3.2e178

   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 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 60.9%

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

   if -9e18 < x < -1.7499999999999999e-51 or 3.2e178 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if -1.7499999999999999e-51 < x < 5

   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 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+18}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+178}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -190:\\ \;\;\;\;\mathsf{fma}\left(z, x, x \cdot y\right)\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -190.0)
   (fma z x (* x y))
   (if (<= x 5.0) (fma z 5.0 (* x y)) (* x (+ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -190.0) {
		tmp = fma(z, x, (x * y));
	} else if (x <= 5.0) {
		tmp = fma(z, 5.0, (x * y));
	} else {
		tmp = x * (z + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -190.0)
		tmp = fma(z, x, Float64(x * y));
	elseif (x <= 5.0)
		tmp = fma(z, 5.0, Float64(x * y));
	else
		tmp = Float64(x * Float64(z + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -190.0], N[(z * x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.0], N[(z * 5.0 + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -190

   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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   if -190 < x < 5

   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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if 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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x \cdot y\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-17}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.75e-51)
   (fma z x (* x y))
   (if (<= x 4.8e-17) (* z 5.0) (* x (+ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.75e-51) {
		tmp = fma(z, x, (x * y));
	} else if (x <= 4.8e-17) {
		tmp = z * 5.0;
	} else {
		tmp = x * (z + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.75e-51)
		tmp = fma(z, x, Float64(x * y));
	elseif (x <= 4.8e-17)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(x * Float64(z + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.75e-51], N[(z * x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-17], N[(z * 5.0), $MachinePrecision], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7499999999999999e-51

   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 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.2%

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

   if -1.7499999999999999e-51 < x < 4.79999999999999973e-17

   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 77.0

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

   5. Applied rewrites 77.0%

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

   if 4.79999999999999973e-17 < 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 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x \cdot y\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-17}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(z + y\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-17}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ z y))))
   (if (<= x -1.75e-51) t_0 (if (<= x 4.8e-17) (* z 5.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (z + y);
	double tmp;
	if (x <= -1.75e-51) {
		tmp = t_0;
	} else if (x <= 4.8e-17) {
		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 * (z + y)
    if (x <= (-1.75d-51)) then
        tmp = t_0
    else if (x <= 4.8d-17) 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 * (z + y);
	double tmp;
	if (x <= -1.75e-51) {
		tmp = t_0;
	} else if (x <= 4.8e-17) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (z + y)
	tmp = 0
	if x <= -1.75e-51:
		tmp = t_0
	elif x <= 4.8e-17:
		tmp = z * 5.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(z + y))
	tmp = 0.0
	if (x <= -1.75e-51)
		tmp = t_0;
	elseif (x <= 4.8e-17)
		tmp = Float64(z * 5.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (z + y);
	tmp = 0.0;
	if (x <= -1.75e-51)
		tmp = t_0;
	elseif (x <= 4.8e-17)
		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[(z + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e-51], t$95$0, If[LessEqual[x, 4.8e-17], N[(z * 5.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7499999999999999e-51 or 4.79999999999999973e-17 < 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 95.4

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

   5. Applied rewrites 95.4%

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

   if -1.7499999999999999e-51 < x < 4.79999999999999973e-17

   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 77.0

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

   5. Applied rewrites 77.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-17}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -60:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -60.0) (* z x) (if (<= x 5.0) (* z 5.0) (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -60.0) {
		tmp = z * x;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else {
		tmp = z * 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 <= (-60.0d0)) then
        tmp = z * x
    else if (x <= 5.0d0) then
        tmp = z * 5.0d0
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -60.0) {
		tmp = z * x;
	} else if (x <= 5.0) {
		tmp = z * 5.0;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -60.0:
		tmp = z * x
	elif x <= 5.0:
		tmp = z * 5.0
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -60.0)
		tmp = Float64(z * x);
	elseif (x <= 5.0)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -60.0)
		tmp = z * x;
	elseif (x <= 5.0)
		tmp = z * 5.0;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -60.0], N[(z * x), $MachinePrecision], If[LessEqual[x, 5.0], N[(z * 5.0), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -60 or 5 < x

   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 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.7%

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

   if -60 < x < 5

   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 72.5

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

   5. Applied rewrites 72.5%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -60:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * x + N[(z * N[(5.0 + x), $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. 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 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 8: 28.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
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 64.3

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 27.5%

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

9. Final simplification 27.5%

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

Developer Target 1: 97.8% 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 2024225 
(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)))