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

Percentage Accurate: 99.9% → 100.0%

Time: 5.0s

Alternatives: 8

Speedup: 1.1×

• 21.0% 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: 60.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+67}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-90}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-53}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+43}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.65e+67)
   (* z x)
   (if (<= x -7e-90)
     (* y x)
     (if (<= x 2.2e-53) (* 5.0 z) (if (<= x 6e+43) (* y x) (* z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.65e+67) {
		tmp = z * x;
	} else if (x <= -7e-90) {
		tmp = y * x;
	} else if (x <= 2.2e-53) {
		tmp = 5.0 * z;
	} else if (x <= 6e+43) {
		tmp = y * x;
	} 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 <= (-2.65d+67)) then
        tmp = z * x
    else if (x <= (-7d-90)) then
        tmp = y * x
    else if (x <= 2.2d-53) then
        tmp = 5.0d0 * z
    else if (x <= 6d+43) then
        tmp = y * x
    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 <= -2.65e+67) {
		tmp = z * x;
	} else if (x <= -7e-90) {
		tmp = y * x;
	} else if (x <= 2.2e-53) {
		tmp = 5.0 * z;
	} else if (x <= 6e+43) {
		tmp = y * x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.65e+67:
		tmp = z * x
	elif x <= -7e-90:
		tmp = y * x
	elif x <= 2.2e-53:
		tmp = 5.0 * z
	elif x <= 6e+43:
		tmp = y * x
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.65e+67)
		tmp = Float64(z * x);
	elseif (x <= -7e-90)
		tmp = Float64(y * x);
	elseif (x <= 2.2e-53)
		tmp = Float64(5.0 * z);
	elseif (x <= 6e+43)
		tmp = Float64(y * x);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.65e+67)
		tmp = z * x;
	elseif (x <= -7e-90)
		tmp = y * x;
	elseif (x <= 2.2e-53)
		tmp = 5.0 * z;
	elseif (x <= 6e+43)
		tmp = y * x;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.65e+67], N[(z * x), $MachinePrecision], If[LessEqual[x, -7e-90], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.2e-53], N[(5.0 * z), $MachinePrecision], If[LessEqual[x, 6e+43], N[(y * x), $MachinePrecision], N[(z * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65e67 or 6.00000000000000033e43 < 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 61.6

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 53.9%

      \[\leadsto \frac{\left(25 - x \cdot x\right) \cdot z}{\color{blue}{5 - x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{25 \cdot z}{5 - x} \]
   9. Step-by-step derivation

   10. Applied rewrites 1.5%

       \[\leadsto \frac{25 \cdot z}{5 - x} \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 61.6%

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

   if -2.65e67 < x < -6.9999999999999997e-90 or 2.20000000000000018e-53 < x < 6.00000000000000033e43

   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 60.9

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

   5. Applied rewrites 60.9%

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

   if -6.9999999999999997e-90 < x < 2.20000000000000018e-53

   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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+67}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-90}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-53}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+43}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 2.4 \cdot 10^{-25}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 2.4e-25))) (* (+ z y) x) (fma 5.0 z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.0) || !(x <= 2.4e-25)) {
		tmp = (z + y) * x;
	} else {
		tmp = fma(5.0, z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 2.4e-25))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = fma(5.0, z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 2.4e-25]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 2.40000000000000009e-25 < 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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 95.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 95.5

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 95.5

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

   6. Applied rewrites 95.5%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if -5 < x < 2.40000000000000009e-25

   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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 2.4 \cdot 10^{-25}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-90} \lor \neg \left(x \leq 1.86 \cdot 10^{-53}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.6e-90) (not (<= x 1.86e-53)))
   (* (+ z y) x)
   (fma z 5.0 (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.6e-90) || !(x <= 1.86e-53)) {
		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 <= -7.6e-90) || !(x <= 1.86e-53))
		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, -7.6e-90], N[Not[LessEqual[x, 1.86e-53]], $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 < -7.6e-90 or 1.8599999999999999e-53 < 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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 96.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 92.7

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

   9. Applied rewrites 92.7%

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

   if -7.6e-90 < x < 1.8599999999999999e-53

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 73.3

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

   7. Applied rewrites 73.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-90} \lor \neg \left(x \leq 1.86 \cdot 10^{-53}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-90} \lor \neg \left(x \leq 1.86 \cdot 10^{-53}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.6e-90) (not (<= x 1.86e-53))) (* (+ z y) x) (* 5.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.6e-90) || !(x <= 1.86e-53)) {
		tmp = (z + y) * x;
	} 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 <= (-7.6d-90)) .or. (.not. (x <= 1.86d-53))) then
        tmp = (z + y) * x
    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 <= -7.6e-90) || !(x <= 1.86e-53)) {
		tmp = (z + y) * x;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.6e-90) or not (x <= 1.86e-53):
		tmp = (z + y) * x
	else:
		tmp = 5.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.6e-90) || !(x <= 1.86e-53))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = Float64(5.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.6e-90) || ~((x <= 1.86e-53)))
		tmp = (z + y) * x;
	else
		tmp = 5.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.6e-90], N[Not[LessEqual[x, 1.86e-53]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.6e-90 or 1.8599999999999999e-53 < 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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-in N/A

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

      13. lift-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 96.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 92.7

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

   9. Applied rewrites 92.7%

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

   if -7.6e-90 < x < 1.8599999999999999e-53

   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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-90} \lor \neg \left(x \leq 1.86 \cdot 10^{-53}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-79} \lor \neg \left(z \leq 7.8 \cdot 10^{-60}\right):\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.45e-79) (not (<= z 7.8e-60))) (* (+ 5.0 x) z) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.45e-79) || !(z <= 7.8e-60)) {
		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 <= (-2.45d-79)) .or. (.not. (z <= 7.8d-60))) 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 <= -2.45e-79) || !(z <= 7.8e-60)) {
		tmp = (5.0 + x) * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.45e-79) or not (z <= 7.8e-60):
		tmp = (5.0 + x) * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.45e-79) || !(z <= 7.8e-60))
		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 <= -2.45e-79) || ~((z <= 7.8e-60)))
		tmp = (5.0 + x) * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.45e-79], N[Not[LessEqual[z, 7.8e-60]], $MachinePrecision]], N[(N[(5.0 + x), $MachinePrecision] * z), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.45e-79 or 7.8000000000000004e-60 < 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 83.6

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

   5. Applied rewrites 83.6%

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

   if -2.45e-79 < z < 7.8000000000000004e-60

   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 69.9

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

   5. Applied rewrites 69.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-79} \lor \neg \left(z \leq 7.8 \cdot 10^{-60}\right):\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0035 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.0035) (not (<= x 5.0))) (* z x) (* 5.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.0035) || !(x <= 5.0)) {
		tmp = z * x;
	} 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 <= (-0.0035d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = z * x
    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 <= -0.0035) || !(x <= 5.0)) {
		tmp = z * x;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.0035) or not (x <= 5.0):
		tmp = z * x
	else:
		tmp = 5.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.0035) || !(x <= 5.0))
		tmp = Float64(z * x);
	else
		tmp = Float64(5.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.0035) || ~((x <= 5.0)))
		tmp = z * x;
	else
		tmp = 5.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.0035], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(z * x), $MachinePrecision], N[(5.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00350000000000000007 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 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 47.7%

      \[\leadsto \frac{\left(25 - x \cdot x\right) \cdot z}{\color{blue}{5 - x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{25 \cdot z}{5 - x} \]
   9. Step-by-step derivation

   10. Applied rewrites 1.8%

       \[\leadsto \frac{25 \cdot z}{5 - x} \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 55.7%

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

   if -0.00350000000000000007 < x < 5

   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 65.3

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

   5. Applied rewrites 65.3%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0035 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. distribute-rgt-in N/A

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

   2. *-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 60.8

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

5. Applied rewrites 60.8%

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

7. Applied rewrites 56.3%

   \[\leadsto \frac{\left(25 - x \cdot x\right) \cdot z}{\color{blue}{5 - x}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{25 \cdot z}{5 - x} \]
9. Step-by-step derivation

10. Applied rewrites 32.8%

    \[\leadsto \frac{25 \cdot z}{5 - x} \]

11. Taylor expanded in x around inf

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

13. Applied rewrites 30.4%

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

14. Final simplification 30.4%

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

Developer Target 1: 98.1% 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 2024326 
(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)))