Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 5

Speedup: 1.2×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-57} \lor \neg \left(y \leq 4.1 \cdot 10^{-108}\right):\\ \;\;\;\;\left(z + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3e-57) (not (<= y 4.1e-108))) (* (+ z x) y) (fma y x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e-57) || !(y <= 4.1e-108)) {
		tmp = (z + x) * y;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3e-57) || !(y <= 4.1e-108))
		tmp = Float64(Float64(z + x) * y);
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3e-57], N[Not[LessEqual[y, 4.1e-108]], $MachinePrecision]], N[(N[(z + x), $MachinePrecision] * y), $MachinePrecision], N[(y * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000001e-57 or 4.10000000000000037e-108 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 90.5

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

   7. Applied rewrites 90.5%

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

   if -3.00000000000000001e-57 < y < 4.10000000000000037e-108

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 72.9

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

   5. Applied rewrites 72.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-57} \lor \neg \left(y \leq 4.1 \cdot 10^{-108}\right):\\ \;\;\;\;\left(z + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-83} \lor \neg \left(x \leq 4.2 \cdot 10^{-85}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e-83) (not (<= x 4.2e-85))) (fma y x x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e-83) || !(x <= 4.2e-85)) {
		tmp = fma(y, x, x);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e-83) || !(x <= 4.2e-85))
		tmp = fma(y, x, x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e-83], N[Not[LessEqual[x, 4.2e-85]], $MachinePrecision]], N[(y * x + x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e-83 or 4.2e-85 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if -5e-83 < x < 4.2e-85

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-83} \lor \neg \left(x \leq 4.2 \cdot 10^{-85}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-25} \lor \neg \left(x \leq 2.4 \cdot 10^{+34}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5e-25) (not (<= x 2.4e+34))) (* y x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e-25) || !(x <= 2.4e+34)) {
		tmp = y * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d-25)) .or. (.not. (x <= 2.4d+34))) then
        tmp = y * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e-25) || !(x <= 2.4e+34)) {
		tmp = y * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.5e-25) or not (x <= 2.4e+34):
		tmp = y * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.5e-25) || !(x <= 2.4e+34))
		tmp = Float64(y * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.5e-25) || ~((x <= 2.4e+34)))
		tmp = y * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.5e-25], N[Not[LessEqual[x, 2.4e+34]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4999999999999999e-25 or 2.39999999999999987e34 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.9%

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

   if -1.4999999999999999e-25 < x < 2.39999999999999987e34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-25} \lor \neg \left(x \leq 2.4 \cdot 10^{+34}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 59.6

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

5. Applied rewrites 59.6%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 29.2%

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

9. Final simplification 29.2%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))