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

Percentage Accurate: 99.9% → 99.9%

Time: 8.4s

Alternatives: 10

Speedup: 1.1×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.0)
   (* x (+ y z))
   (if (<= x 2.7e-6) (fma z 5.0 (* y x)) (fma z x (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.0) {
		tmp = x * (y + z);
	} else if (x <= 2.7e-6) {
		tmp = fma(z, 5.0, (y * x));
	} else {
		tmp = fma(z, x, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(x * Float64(y + z));
	elseif (x <= 2.7e-6)
		tmp = fma(z, 5.0, Float64(y * x));
	else
		tmp = fma(z, x, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.0], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-6], N[(z * 5.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * x + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -5 < x < 2.69999999999999998e-6

   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. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.8%

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

   if 2.69999999999999998e-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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.0)
   (* x (+ y z))
   (if (<= x 2.7e-6) (fma y x (* z 5.0)) (fma z x (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.0) {
		tmp = x * (y + z);
	} else if (x <= 2.7e-6) {
		tmp = fma(y, x, (z * 5.0));
	} else {
		tmp = fma(z, x, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(x * Float64(y + z));
	elseif (x <= 2.7e-6)
		tmp = fma(y, x, Float64(z * 5.0));
	else
		tmp = fma(z, x, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.0], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-6], N[(y * x + N[(z * 5.0), $MachinePrecision]), $MachinePrecision], N[(z * x + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -5 < x < 2.69999999999999998e-6

   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. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 98.8

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

   9. Applied rewrites 98.8%

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

   if 2.69999999999999998e-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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e-49)
   (* x (+ y z))
   (if (<= x 1.6e-100) (* z (+ x 5.0)) (fma z x (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-49) {
		tmp = x * (y + z);
	} else if (x <= 1.6e-100) {
		tmp = z * (x + 5.0);
	} else {
		tmp = fma(z, x, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e-49)
		tmp = Float64(x * Float64(y + z));
	elseif (x <= 1.6e-100)
		tmp = Float64(z * Float64(x + 5.0));
	else
		tmp = fma(z, x, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e-49], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-100], N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision], N[(z * x + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e-49

   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 91.5

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

   5. Applied rewrites 91.5%

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

   if -6.2e-49 < x < 1.60000000000000008e-100

   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 72.1

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

   5. Applied rewrites 72.1%

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

   if 1.60000000000000008e-100 < x

   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 89.8

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

   5. Applied rewrites 89.8%

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

   7. Applied rewrites 89.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))))
   (if (<= x -6.2e-49) t_0 (if (<= x 1.6e-100) (* z (+ x 5.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -6.2e-49) {
		tmp = t_0;
	} else if (x <= 1.6e-100) {
		tmp = z * (x + 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + z)
    if (x <= (-6.2d-49)) then
        tmp = t_0
    else if (x <= 1.6d-100) then
        tmp = z * (x + 5.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -6.2e-49) {
		tmp = t_0;
	} else if (x <= 1.6e-100) {
		tmp = z * (x + 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y + z)
	tmp = 0
	if x <= -6.2e-49:
		tmp = t_0
	elif x <= 1.6e-100:
		tmp = z * (x + 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + z))
	tmp = 0.0
	if (x <= -6.2e-49)
		tmp = t_0;
	elseif (x <= 1.6e-100)
		tmp = Float64(z * Float64(x + 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y + z);
	tmp = 0.0;
	if (x <= -6.2e-49)
		tmp = t_0;
	elseif (x <= 1.6e-100)
		tmp = z * (x + 5.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-49], t$95$0, If[LessEqual[x, 1.6e-100], N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-49 or 1.60000000000000008e-100 < x

   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 90.5

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

   5. Applied rewrites 90.5%

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

   if -6.2e-49 < x < 1.60000000000000008e-100

   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 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + z\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ y z))))
   (if (<= x -6.2e-49) t_0 (if (<= x 1.6e-100) (* z 5.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -6.2e-49) {
		tmp = t_0;
	} else if (x <= 1.6e-100) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y + z)
    if (x <= (-6.2d-49)) then
        tmp = t_0
    else if (x <= 1.6d-100) then
        tmp = z * 5.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y + z);
	double tmp;
	if (x <= -6.2e-49) {
		tmp = t_0;
	} else if (x <= 1.6e-100) {
		tmp = z * 5.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y + z)
	tmp = 0
	if x <= -6.2e-49:
		tmp = t_0
	elif x <= 1.6e-100:
		tmp = z * 5.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y + z))
	tmp = 0.0
	if (x <= -6.2e-49)
		tmp = t_0;
	elseif (x <= 1.6e-100)
		tmp = Float64(z * 5.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y + z);
	tmp = 0.0;
	if (x <= -6.2e-49)
		tmp = t_0;
	elseif (x <= 1.6e-100)
		tmp = z * 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-49], t$95$0, If[LessEqual[x, 1.6e-100], N[(z * 5.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-49 or 1.60000000000000008e-100 < x

   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 90.5

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

   5. Applied rewrites 90.5%

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

   if -6.2e-49 < x < 1.60000000000000008e-100

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-100}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{-49}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.68 \cdot 10^{-100}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.46e-49) (* y x) (if (<= x 1.68e-100) (* z 5.0) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.46e-49) {
		tmp = y * x;
	} else if (x <= 1.68e-100) {
		tmp = z * 5.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.46d-49)) then
        tmp = y * x
    else if (x <= 1.68d-100) then
        tmp = z * 5.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.46e-49) {
		tmp = y * x;
	} else if (x <= 1.68e-100) {
		tmp = z * 5.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.46e-49:
		tmp = y * x
	elif x <= 1.68e-100:
		tmp = z * 5.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.46e-49)
		tmp = Float64(y * x);
	elseif (x <= 1.68e-100)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.46e-49)
		tmp = y * x;
	elseif (x <= 1.68e-100)
		tmp = z * 5.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.46e-49], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.68e-100], N[(z * 5.0), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.46000000000000007e-49 or 1.68000000000000012e-100 < 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 62.2

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

   5. Applied rewrites 62.2%

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

   if -1.46000000000000007e-49 < x < 1.68000000000000012e-100

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{-49}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.68 \cdot 10^{-100}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 0.00325:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.0) (* z x) (if (<= x 0.00325) (* z 5.0) (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.0) {
		tmp = z * x;
	} else if (x <= 0.00325) {
		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 <= (-5.0d0)) then
        tmp = z * x
    else if (x <= 0.00325d0) 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 <= -5.0) {
		tmp = z * x;
	} else if (x <= 0.00325) {
		tmp = z * 5.0;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.0:
		tmp = z * x
	elif x <= 0.00325:
		tmp = z * 5.0
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(z * x);
	elseif (x <= 0.00325)
		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 <= -5.0)
		tmp = z * x;
	elseif (x <= 0.00325)
		tmp = z * 5.0;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.0], N[(z * x), $MachinePrecision], If[LessEqual[x, 0.00325], N[(z * 5.0), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 0.00324999999999999985 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 41.1%

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

   if -5 < x < 0.00324999999999999985

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 0.00325:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-out N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 10: 27.6% 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 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 68.6

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

5. Applied rewrites 68.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 22.6%

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

Developer Target 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024233 
(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)))