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

Percentage Accurate: 99.9% → 100.0%

Time: 7.6s

Alternatives: 8

Speedup: 1.0×

• 20.7% 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, 0.1× 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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 61.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+192}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -920000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-14}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+179}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e+192)
   (* x y)
   (if (<= x -920000000.0)
     (* z x)
     (if (<= x -1.3e-72)
       (* x y)
       (if (<= x 1.02e-14)
         (* z 5.0)
         (if (<= x 8.5e+95) (* x y) (if (<= x 2.6e+179) (* z x) (* x y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e+192) {
		tmp = x * y;
	} else if (x <= -920000000.0) {
		tmp = z * x;
	} else if (x <= -1.3e-72) {
		tmp = x * y;
	} else if (x <= 1.02e-14) {
		tmp = z * 5.0;
	} else if (x <= 8.5e+95) {
		tmp = x * y;
	} else if (x <= 2.6e+179) {
		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 <= (-3.9d+192)) then
        tmp = x * y
    else if (x <= (-920000000.0d0)) then
        tmp = z * x
    else if (x <= (-1.3d-72)) then
        tmp = x * y
    else if (x <= 1.02d-14) then
        tmp = z * 5.0d0
    else if (x <= 8.5d+95) then
        tmp = x * y
    else if (x <= 2.6d+179) 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 <= -3.9e+192) {
		tmp = x * y;
	} else if (x <= -920000000.0) {
		tmp = z * x;
	} else if (x <= -1.3e-72) {
		tmp = x * y;
	} else if (x <= 1.02e-14) {
		tmp = z * 5.0;
	} else if (x <= 8.5e+95) {
		tmp = x * y;
	} else if (x <= 2.6e+179) {
		tmp = z * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.9e+192:
		tmp = x * y
	elif x <= -920000000.0:
		tmp = z * x
	elif x <= -1.3e-72:
		tmp = x * y
	elif x <= 1.02e-14:
		tmp = z * 5.0
	elif x <= 8.5e+95:
		tmp = x * y
	elif x <= 2.6e+179:
		tmp = z * x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e+192)
		tmp = Float64(x * y);
	elseif (x <= -920000000.0)
		tmp = Float64(z * x);
	elseif (x <= -1.3e-72)
		tmp = Float64(x * y);
	elseif (x <= 1.02e-14)
		tmp = Float64(z * 5.0);
	elseif (x <= 8.5e+95)
		tmp = Float64(x * y);
	elseif (x <= 2.6e+179)
		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 <= -3.9e+192)
		tmp = x * y;
	elseif (x <= -920000000.0)
		tmp = z * x;
	elseif (x <= -1.3e-72)
		tmp = x * y;
	elseif (x <= 1.02e-14)
		tmp = z * 5.0;
	elseif (x <= 8.5e+95)
		tmp = x * y;
	elseif (x <= 2.6e+179)
		tmp = z * x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e+192], N[(x * y), $MachinePrecision], If[LessEqual[x, -920000000.0], N[(z * x), $MachinePrecision], If[LessEqual[x, -1.3e-72], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.02e-14], N[(z * 5.0), $MachinePrecision], If[LessEqual[x, 8.5e+95], N[(x * y), $MachinePrecision], If[LessEqual[x, 2.6e+179], N[(z * x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8999999999999998e192 or -9.2e8 < x < -1.29999999999999998e-72 or 1.02e-14 < x < 8.5000000000000002e95 or 2.6000000000000002e179 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 67.5%

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

   if -3.8999999999999998e192 < x < -9.2e8 or 8.5000000000000002e95 < x < 2.6000000000000002e179

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.2%

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

      1. +-commutative 98.2%

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

   4. Simplified 98.2%

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

   5. Taylor expanded in z around inf 63.8%

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

   if -1.29999999999999998e-72 < x < 1.02e-14

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+192}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -920000000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-14}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+95}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+179}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45000 \lor \neg \left(x \leq 0.057\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z \cdot -5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -45000.0) (not (<= x 0.057)))
   (* x (+ z y))
   (- (* x y) (* z -5.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -45000.0) || !(x <= 0.057)) {
		tmp = x * (z + y);
	} else {
		tmp = (x * y) - (z * -5.0);
	}
	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) :: tmp
    if ((x <= (-45000.0d0)) .or. (.not. (x <= 0.057d0))) then
        tmp = x * (z + y)
    else
        tmp = (x * y) - (z * (-5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -45000.0) || !(x <= 0.057)) {
		tmp = x * (z + y);
	} else {
		tmp = (x * y) - (z * -5.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -45000.0) or not (x <= 0.057):
		tmp = x * (z + y)
	else:
		tmp = (x * y) - (z * -5.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -45000.0) || !(x <= 0.057))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(Float64(x * y) - Float64(z * -5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -45000.0) || ~((x <= 0.057)))
		tmp = x * (z + y);
	else
		tmp = (x * y) - (z * -5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -45000.0], N[Not[LessEqual[x, 0.057]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - N[(z * -5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -45000 or 0.0570000000000000021 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.6%

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

      1. +-commutative 98.6%

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

   4. Simplified 98.6%

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

   if -45000 < x < 0.0570000000000000021

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around -inf 99.9%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(-1 \cdot x - 5\right)\right) + x \cdot y} \]
   5. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

         \[\leadsto \color{blue}{x \cdot y - z \cdot \left(-1 \cdot x - 5\right)} \]

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. mul-1-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -45000 \lor \neg \left(x \leq 0.057\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z \cdot -5\\ \end{array} \]

Alternative 4: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-66} \lor \neg \left(x \leq 5.2 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.95e-66) (not (<= x 5.2e-14))) (* x (+ z y)) (* z 5.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e-66) || !(x <= 5.2e-14)) {
		tmp = x * (z + y);
	} else {
		tmp = z * 5.0;
	}
	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) :: tmp
    if ((x <= (-1.95d-66)) .or. (.not. (x <= 5.2d-14))) then
        tmp = x * (z + y)
    else
        tmp = z * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.95e-66) || !(x <= 5.2e-14)) {
		tmp = x * (z + y);
	} else {
		tmp = z * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.95e-66) or not (x <= 5.2e-14):
		tmp = x * (z + y)
	else:
		tmp = z * 5.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.95e-66) || !(x <= 5.2e-14))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(z * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.95e-66) || ~((x <= 5.2e-14)))
		tmp = x * (z + y);
	else
		tmp = z * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.95e-66], N[Not[LessEqual[x, 5.2e-14]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999991e-66 or 5.19999999999999993e-14 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.4%

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

      1. +-commutative 95.4%

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

   4. Simplified 95.4%

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

   if -1.94999999999999991e-66 < x < 5.19999999999999993e-14

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-66} \lor \neg \left(x \leq 5.2 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot 5\\ \end{array} \]

Alternative 5: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-66} \lor \neg \left(x \leq 0.018\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.2e-66) (not (<= x 0.018))) (* x (+ z y)) (* z (+ 5.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.2e-66) || !(x <= 0.018)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + 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 <= (-2.2d-66)) .or. (.not. (x <= 0.018d0))) then
        tmp = x * (z + y)
    else
        tmp = z * (5.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.2e-66) || !(x <= 0.018)) {
		tmp = x * (z + y);
	} else {
		tmp = z * (5.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.2e-66) or not (x <= 0.018):
		tmp = x * (z + y)
	else:
		tmp = z * (5.0 + x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.2e-66) || !(x <= 0.018))
		tmp = Float64(x * Float64(z + y));
	else
		tmp = Float64(z * Float64(5.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.2e-66) || ~((x <= 0.018)))
		tmp = x * (z + y);
	else
		tmp = z * (5.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.2e-66], N[Not[LessEqual[x, 0.018]], $MachinePrecision]], N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision], N[(z * N[(5.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000001e-66 or 0.0179999999999999986 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 96.6%

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

      1. +-commutative 96.6%

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

   4. Simplified 96.6%

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

   if -2.2000000000000001e-66 < x < 0.0179999999999999986

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 77.6%

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

      1. distribute-rgt-in 77.6%

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

   4. Simplified 77.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-66} \lor \neg \left(x \leq 0.018\right):\\ \;\;\;\;x \cdot \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(5 + x\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * (z + y)) + (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 * (z + y)) + (z * 5.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 7: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-76}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.8e-76) (* x y) (if (<= x 1.8e-13) (* z 5.0) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.8e-76) {
		tmp = x * y;
	} else if (x <= 1.8e-13) {
		tmp = z * 5.0;
	} 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 <= (-9.8d-76)) then
        tmp = x * y
    else if (x <= 1.8d-13) then
        tmp = z * 5.0d0
    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 <= -9.8e-76) {
		tmp = x * y;
	} else if (x <= 1.8e-13) {
		tmp = z * 5.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.8e-76:
		tmp = x * y
	elif x <= 1.8e-13:
		tmp = z * 5.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.8e-76)
		tmp = Float64(x * y);
	elseif (x <= 1.8e-13)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.8e-76)
		tmp = x * y;
	elseif (x <= 1.8e-13)
		tmp = z * 5.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.8e-76], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.8e-13], N[(z * 5.0), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.79999999999999944e-76 or 1.7999999999999999e-13 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 54.2%

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

   if -9.79999999999999944e-76 < x < 1.7999999999999999e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 78.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-76}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 36.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ z \cdot 5 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = z * 5.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 39.1%

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

3. Final simplification 39.1%

   \[\leadsto z \cdot 5 \]

Developer target: 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 2023279 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :herbie-target
  (+ (* (+ x 5.0) z) (* x y))

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