Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4

Percentage Accurate: 99.9% → 100.0%

Time: 11.5s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + 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 = ((((x + y) + y) + x) + z) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((((x + y) + y) + x) + 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 = ((((x + y) + y) + x) + z) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 3, \mathsf{fma}\left(2, y, z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x 3.0 (fma 2.0 y z)))

C

double code(double x, double y, double z) {
	return fma(x, 3.0, fma(2.0, y, z));
}

Julia

function code(x, y, z)
	return fma(x, 3.0, fma(2.0, y, z))
end

Wolfram

code[x_, y_, z_] := N[(x * 3.0 + N[(2.0 * y + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-+ N/A

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. associate-+l+ N/A

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

   10. +-commutative N/A

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

   11. count-2 N/A

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

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{x + \color{blue}{\mathsf{fma}\left(2, x + y, z\right)}}} \]

   13. +-lowering-+.f64 99.7

       \[\leadsto \frac{1}{\frac{1}{x + \mathsf{fma}\left(2, \color{blue}{x + y}, z\right)}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{x + \mathsf{fma}\left(2, x + y, z\right)}}} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{z + 2 \cdot y}{x} - 3\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{z + 2 \cdot y}{x} - 3\right)\right)} \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{z + 2 \cdot y}{x} - 3\right)\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{z + 2 \cdot y}{x} - 3\right)\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{z + 2 \cdot y}{x} + \left(\mathsf{neg}\left(3\right)\right)\right)}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{z + 2 \cdot y}{x} + \color{blue}{-3}\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-3 + -1 \cdot \frac{z + 2 \cdot y}{x}\right)}\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-3\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{z + 2 \cdot y}{x}\right)\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto x \cdot \left(\color{blue}{3} + \left(\mathsf{neg}\left(-1 \cdot \frac{z + 2 \cdot y}{x}\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto x \cdot \left(3 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{z + 2 \cdot y}{x}\right)\right)}\right)\right)\right) \]

   10. distribute-neg-frac2 N/A

       \[\leadsto x \cdot \left(3 + \left(\mathsf{neg}\left(\color{blue}{\frac{z + 2 \cdot y}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto x \cdot \left(3 + \left(\mathsf{neg}\left(\frac{z + 2 \cdot y}{\color{blue}{-1 \cdot x}}\right)\right)\right) \]

   12. distribute-frac-neg2 N/A

       \[\leadsto x \cdot \left(3 + \color{blue}{\frac{z + 2 \cdot y}{\mathsf{neg}\left(-1 \cdot x\right)}}\right) \]

   13. mul-1-neg N/A

       \[\leadsto x \cdot \left(3 + \frac{z + 2 \cdot y}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}\right) \]

   14. remove-double-neg N/A

       \[\leadsto x \cdot \left(3 + \frac{z + 2 \cdot y}{\color{blue}{x}}\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto x \cdot \color{blue}{\left(3 + \frac{z + 2 \cdot y}{x}\right)} \]

   16. /-lowering-/.f64 N/A

       \[\leadsto x \cdot \left(3 + \color{blue}{\frac{z + 2 \cdot y}{x}}\right) \]

   17. +-commutative N/A

       \[\leadsto x \cdot \left(3 + \frac{\color{blue}{2 \cdot y + z}}{x}\right) \]

   18. accelerator-lowering-fma.f64 82.0

       \[\leadsto x \cdot \left(3 + \frac{\color{blue}{\mathsf{fma}\left(2, y, z\right)}}{x}\right) \]

7. Simplified 82.0%

   \[\leadsto \color{blue}{x \cdot \left(3 + \frac{\mathsf{fma}\left(2, y, z\right)}{x}\right)} \]
8. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot 3 + x \cdot \frac{2 \cdot y + z}{x}} \]

   2. *-commutative N/A

      \[\leadsto x \cdot 3 + \color{blue}{\frac{2 \cdot y + z}{x} \cdot x} \]

   3. div-inv N/A

      \[\leadsto x \cdot 3 + \color{blue}{\left(\left(2 \cdot y + z\right) \cdot \frac{1}{x}\right)} \cdot x \]

   4. associate-*l* N/A

      \[\leadsto x \cdot 3 + \color{blue}{\left(2 \cdot y + z\right) \cdot \left(\frac{1}{x} \cdot x\right)} \]

   5. lft-mult-inverse N/A

      \[\leadsto x \cdot 3 + \left(2 \cdot y + z\right) \cdot \color{blue}{1} \]

   6. *-rgt-identity N/A

      \[\leadsto x \cdot 3 + \color{blue}{\left(2 \cdot y + z\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, 2 \cdot y + z\right)} \]

   8. accelerator-lowering-fma.f64 100.0

      \[\leadsto \mathsf{fma}\left(x, 3, \color{blue}{\mathsf{fma}\left(2, y, z\right)}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, \mathsf{fma}\left(2, y, z\right)\right)} \]
10. Add Preprocessing

Alternative 2: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.1e+132)
   (fma x 3.0 z)
   (if (<= z 1.4e+133) (fma 2.0 (+ x y) x) (fma 2.0 y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.1e+132) {
		tmp = fma(x, 3.0, z);
	} else if (z <= 1.4e+133) {
		tmp = fma(2.0, (x + y), x);
	} else {
		tmp = fma(2.0, y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.1e+132)
		tmp = fma(x, 3.0, z);
	elseif (z <= 1.4e+133)
		tmp = fma(2.0, Float64(x + y), x);
	else
		tmp = fma(2.0, y, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.1e+132], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[z, 1.4e+133], N[(2.0 * N[(x + y), $MachinePrecision] + x), $MachinePrecision], N[(2.0 * y + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999992e132

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot x + z \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 3} + z \]

      6. accelerator-lowering-fma.f64 84.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, z\right)} \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, z\right)} \]

   if -4.09999999999999992e132 < z < 1.40000000000000008e133

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 93.3

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

   5. Simplified 93.3%

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

   if 1.40000000000000008e133 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z + 2 \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot y + z} \]

      2. accelerator-lowering-fma.f64 94.5

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

   5. Simplified 94.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(2, x + y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \mathbf{elif}\;x \leq 90000000:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 3, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.2e+46)
   (fma x 3.0 z)
   (if (<= x 90000000.0) (fma 2.0 y z) (fma x 3.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e+46) {
		tmp = fma(x, 3.0, z);
	} else if (x <= 90000000.0) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = fma(x, 3.0, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.2e+46)
		tmp = fma(x, 3.0, z);
	elseif (x <= 90000000.0)
		tmp = fma(2.0, y, z);
	else
		tmp = fma(x, 3.0, z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e+46], N[(x * 3.0 + z), $MachinePrecision], If[LessEqual[x, 90000000.0], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0 + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999997e46 or 9e7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot x + z \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 3} + z \]

      6. accelerator-lowering-fma.f64 83.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, z\right)} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3, z\right)} \]

   if -7.1999999999999997e46 < x < 9e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z + 2 \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot y + z} \]

      2. accelerator-lowering-fma.f64 92.7

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

   5. Simplified 92.7%

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

Alternative 4: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+127}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(2, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+127) (* x 3.0) (if (<= x 7e+119) (fma 2.0 y z) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+127) {
		tmp = x * 3.0;
	} else if (x <= 7e+119) {
		tmp = fma(2.0, y, z);
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e+127)
		tmp = Float64(x * 3.0);
	elseif (x <= 7e+119)
		tmp = fma(2.0, y, z);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e+127], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 7e+119], N[(2.0 * y + z), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e127 or 7.0000000000000001e119 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 3} \]

      2. *-lowering-*.f64 85.2

         \[\leadsto \color{blue}{x \cdot 3} \]

   5. Simplified 85.2%

      \[\leadsto \color{blue}{x \cdot 3} \]

   if -1.2000000000000001e127 < x < 7.0000000000000001e119

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z + 2 \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot y + z} \]

      2. accelerator-lowering-fma.f64 82.9

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

   5. Simplified 82.9%

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

Alternative 5: 51.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(2, y, x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+163}:\\ \;\;\;\;x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e-83)
   (fma 2.0 y x)
   (if (<= y 2.2e+163) (* x 3.0) (fma 2.0 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e-83) {
		tmp = fma(2.0, y, x);
	} else if (y <= 2.2e+163) {
		tmp = x * 3.0;
	} else {
		tmp = fma(2.0, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e-83)
		tmp = fma(2.0, y, x);
	elseif (y <= 2.2e+163)
		tmp = Float64(x * 3.0);
	else
		tmp = fma(2.0, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e-83], N[(2.0 * y + x), $MachinePrecision], If[LessEqual[y, 2.2e+163], N[(x * 3.0), $MachinePrecision], N[(2.0 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4999999999999997e-83 or 2.19999999999999986e163 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{y}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 72.1%

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

   if -7.4999999999999997e-83 < y < 2.19999999999999986e163

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 3} \]

      2. *-lowering-*.f64 61.3

         \[\leadsto \color{blue}{x \cdot 3} \]

   5. Simplified 61.3%

      \[\leadsto \color{blue}{x \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+37}:\\ \;\;\;\;x \cdot 3\\ \mathbf{elif}\;x \leq 112000000:\\ \;\;\;\;2 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+37) (* x 3.0) (if (<= x 112000000.0) (* 2.0 y) (* x 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+37) {
		tmp = x * 3.0;
	} else if (x <= 112000000.0) {
		tmp = 2.0 * y;
	} else {
		tmp = x * 3.0;
	}
	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 <= (-1.8d+37)) then
        tmp = x * 3.0d0
    else if (x <= 112000000.0d0) then
        tmp = 2.0d0 * y
    else
        tmp = x * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+37) {
		tmp = x * 3.0;
	} else if (x <= 112000000.0) {
		tmp = 2.0 * y;
	} else {
		tmp = x * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.8e+37:
		tmp = x * 3.0
	elif x <= 112000000.0:
		tmp = 2.0 * y
	else:
		tmp = x * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+37)
		tmp = Float64(x * 3.0);
	elseif (x <= 112000000.0)
		tmp = Float64(2.0 * y);
	else
		tmp = Float64(x * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e+37)
		tmp = x * 3.0;
	elseif (x <= 112000000.0)
		tmp = 2.0 * y;
	else
		tmp = x * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+37], N[(x * 3.0), $MachinePrecision], If[LessEqual[x, 112000000.0], N[(2.0 * y), $MachinePrecision], N[(x * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.79999999999999999e37 or 1.12e8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot 3} \]

      2. *-lowering-*.f64 71.8

         \[\leadsto \color{blue}{x \cdot 3} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{x \cdot 3} \]

   if -1.79999999999999999e37 < x < 1.12e8

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{x + \color{blue}{\mathsf{fma}\left(2, x + y, z\right)}}} \]

      13. +-lowering-+.f64 99.8

          \[\leadsto \frac{1}{\frac{1}{x + \mathsf{fma}\left(2, \color{blue}{x + y}, z\right)}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x + \mathsf{fma}\left(2, x + y, z\right)}}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 58.2

         \[\leadsto \color{blue}{2 \cdot y} \]

   7. Simplified 58.2%

      \[\leadsto \color{blue}{2 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+116}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e-83) (* 2.0 y) (if (<= y 1.95e+116) (+ x z) (* 2.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e-83) {
		tmp = 2.0 * y;
	} else if (y <= 1.95e+116) {
		tmp = x + z;
	} else {
		tmp = 2.0 * 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 (y <= (-7.5d-83)) then
        tmp = 2.0d0 * y
    else if (y <= 1.95d+116) then
        tmp = x + z
    else
        tmp = 2.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e-83) {
		tmp = 2.0 * y;
	} else if (y <= 1.95e+116) {
		tmp = x + z;
	} else {
		tmp = 2.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e-83:
		tmp = 2.0 * y
	elif y <= 1.95e+116:
		tmp = x + z
	else:
		tmp = 2.0 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e-83)
		tmp = Float64(2.0 * y);
	elseif (y <= 1.95e+116)
		tmp = Float64(x + z);
	else
		tmp = Float64(2.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.5e-83)
		tmp = 2.0 * y;
	elseif (y <= 1.95e+116)
		tmp = x + z;
	else
		tmp = 2.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e-83], N[(2.0 * y), $MachinePrecision], If[LessEqual[y, 1.95e+116], N[(x + z), $MachinePrecision], N[(2.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4999999999999997e-83 or 1.95000000000000016e116 < y

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. count-2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{x + \color{blue}{\mathsf{fma}\left(2, x + y, z\right)}}} \]

      13. +-lowering-+.f64 99.7

          \[\leadsto \frac{1}{\frac{1}{x + \mathsf{fma}\left(2, \color{blue}{x + y}, z\right)}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x + \mathsf{fma}\left(2, x + y, z\right)}}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 66.0

         \[\leadsto \color{blue}{2 \cdot y} \]

   7. Simplified 66.0%

      \[\leadsto \color{blue}{2 \cdot y} \]

   if -7.4999999999999997e-83 < y < 1.95000000000000016e116

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 43.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+116}:\\ \;\;\;\;x + z\\ \mathbf{else}:\\ \;\;\;\;2 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. count-2 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 9: 34.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+146}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x 3e+146) z x))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+146) {
		tmp = z;
	} else {
		tmp = 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 <= 3d+146) then
        tmp = z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+146) {
		tmp = z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3e+146:
		tmp = z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e+146)
		tmp = z;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3e+146)
		tmp = z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e+146], z, x]

Derivation

1. Split input into 2 regimes

2. if x < 3.00000000000000002e146

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 28.4%

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

   if 3.00000000000000002e146 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 17.3%

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

   6. Taylor expanded in z around 0

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

   8. Simplified 17.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 39.1% accurate, 4.0× speedup

Math

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

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%

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

3. Taylor expanded in z around inf

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

5. Simplified 30.6%

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

6. Final simplification 30.6%

   \[\leadsto x + z \]
7. Add Preprocessing

Alternative 11: 8.0% accurate, 16.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

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

5. Simplified 30.6%

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

6. Taylor expanded in z around 0

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

8. Simplified 9.4%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendInside from plot-0.2.3.4"
  :precision binary64
  (+ (+ (+ (+ (+ x y) y) x) z) x))