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

Percentage Accurate: 99.9% → 100.0%

Time: 6.0s

Alternatives: 9

Speedup: 1.1×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 61.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+251}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+251)
   (* y x)
   (if (<= x -9.5e+36)
     (* x z)
     (if (<= x -7.5e-11) (* y x) (if (<= x 5.0) (* 5.0 z) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+251) {
		tmp = y * x;
	} else if (x <= -9.5e+36) {
		tmp = x * z;
	} else if (x <= -7.5e-11) {
		tmp = y * x;
	} else if (x <= 5.0) {
		tmp = 5.0 * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1d+251)) then
        tmp = y * x
    else if (x <= (-9.5d+36)) then
        tmp = x * z
    else if (x <= (-7.5d-11)) then
        tmp = y * x
    else if (x <= 5.0d0) then
        tmp = 5.0d0 * z
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+251) {
		tmp = y * x;
	} else if (x <= -9.5e+36) {
		tmp = x * z;
	} else if (x <= -7.5e-11) {
		tmp = y * x;
	} else if (x <= 5.0) {
		tmp = 5.0 * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e+251:
		tmp = y * x
	elif x <= -9.5e+36:
		tmp = x * z
	elif x <= -7.5e-11:
		tmp = y * x
	elif x <= 5.0:
		tmp = 5.0 * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+251)
		tmp = Float64(y * x);
	elseif (x <= -9.5e+36)
		tmp = Float64(x * z);
	elseif (x <= -7.5e-11)
		tmp = Float64(y * x);
	elseif (x <= 5.0)
		tmp = Float64(5.0 * z);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e+251)
		tmp = y * x;
	elseif (x <= -9.5e+36)
		tmp = x * z;
	elseif (x <= -7.5e-11)
		tmp = y * x;
	elseif (x <= 5.0)
		tmp = 5.0 * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e+251], N[(y * x), $MachinePrecision], If[LessEqual[x, -9.5e+36], N[(x * z), $MachinePrecision], If[LessEqual[x, -7.5e-11], N[(y * x), $MachinePrecision], If[LessEqual[x, 5.0], N[(5.0 * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e251 or -9.49999999999999974e36 < x < -7.5e-11

   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. *-commutative N/A

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

      2. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -1e251 < x < -9.49999999999999974e36 or 5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.7%

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

   if -7.5e-11 < x < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+251}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 5:\\ \;\;\;\;5 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -34000000 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 5 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -34000000.0) (not (<= x 5.0)))
   (* (+ z y) x)
   (fma y x (* 5.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -34000000.0) || !(x <= 5.0)) {
		tmp = (z + y) * x;
	} else {
		tmp = fma(y, x, (5.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -34000000.0) || !(x <= 5.0))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = fma(y, x, Float64(5.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -34000000.0], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(y * x + N[(5.0 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e7 or 5 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -3.4e7 < x < 5

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -34000000 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 5 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 2.3 \cdot 10^{-8}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.084) (not (<= x 2.3e-8))) (* (+ z y) x) (fma z 5.0 (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.084) || !(x <= 2.3e-8)) {
		tmp = (z + y) * x;
	} else {
		tmp = fma(z, 5.0, (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.084) || !(x <= 2.3e-8))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = fma(z, 5.0, Float64(x * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.084], N[Not[LessEqual[x, 2.3e-8]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(z * 5.0 + N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0840000000000000052 or 2.3000000000000001e-8 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -0.0840000000000000052 < x < 2.3000000000000001e-8

   1. Initial program 99.8%

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

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 77.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 2.3 \cdot 10^{-8}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 2.3 \cdot 10^{-8}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.084) (not (<= x 2.3e-8))) (* (+ z y) x) (* (+ 5.0 x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.084) || !(x <= 2.3e-8)) {
		tmp = (z + y) * x;
	} else {
		tmp = (5.0 + x) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-0.084d0)) .or. (.not. (x <= 2.3d-8))) then
        tmp = (z + y) * x
    else
        tmp = (5.0d0 + x) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.084) || !(x <= 2.3e-8)) {
		tmp = (z + y) * x;
	} else {
		tmp = (5.0 + x) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.084) or not (x <= 2.3e-8):
		tmp = (z + y) * x
	else:
		tmp = (5.0 + x) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.084) || !(x <= 2.3e-8))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = Float64(Float64(5.0 + x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.084) || ~((x <= 2.3e-8)))
		tmp = (z + y) * x;
	else
		tmp = (5.0 + x) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.084], N[Not[LessEqual[x, 2.3e-8]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(N[(5.0 + x), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0840000000000000052 or 2.3000000000000001e-8 < x

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -0.0840000000000000052 < x < 2.3000000000000001e-8

   1. Initial program 99.8%

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

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 77.7

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

   5. Applied rewrites 77.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 2.3 \cdot 10^{-8}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-87} \lor \neg \left(z \leq 4.35 \cdot 10^{-156}\right):\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.7e-87) (not (<= z 4.35e-156))) (* (+ 5.0 x) z) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.7e-87) || !(z <= 4.35e-156)) {
		tmp = (5.0 + x) * z;
	} 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 ((z <= (-2.7d-87)) .or. (.not. (z <= 4.35d-156))) then
        tmp = (5.0d0 + x) * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.7e-87) || !(z <= 4.35e-156)) {
		tmp = (5.0 + x) * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.7e-87) or not (z <= 4.35e-156):
		tmp = (5.0 + x) * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.7e-87) || !(z <= 4.35e-156))
		tmp = Float64(Float64(5.0 + x) * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.7e-87) || ~((z <= 4.35e-156)))
		tmp = (5.0 + x) * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.7e-87], N[Not[LessEqual[z, 4.35e-156]], $MachinePrecision]], N[(N[(5.0 + x), $MachinePrecision] * z), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.69999999999999984e-87 or 4.35000000000000005e-156 < z

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -2.69999999999999984e-87 < z < 4.35000000000000005e-156

   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. *-commutative N/A

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

      2. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-87} \lor \neg \left(z \leq 4.35 \cdot 10^{-156}\right):\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-5.0d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = x * z
    else
        tmp = 5.0d0 * z
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -5 or 5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.8%

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

   if -5 < x < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. lift-*.f64 N/A

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

   9. associate-+r+ N/A

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

   10. distribute-lft-in N/A

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

   11. lift-+.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 9: 28.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in 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. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 67.8

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 30.4%

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

9. Final simplification 30.4%

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

Developer Target 1: 98.0% 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 2024338 
(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)))