Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

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: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.003 \lor \neg \left(y \leq 1.28 \cdot 10^{-102}\right):\\ \;\;\;\;\left(z + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.003) (not (<= y 1.28e-102))) (* (+ z x) y) (fma y x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.003) || !(y <= 1.28e-102)) {
		tmp = (z + x) * y;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.003) || !(y <= 1.28e-102))
		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, -0.003], N[Not[LessEqual[y, 1.28e-102]], $MachinePrecision]], N[(N[(z + x), $MachinePrecision] * y), $MachinePrecision], N[(y * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0030000000000000001 or 1.27999999999999991e-102 < y

   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 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -0.0030000000000000001 < y < 1.27999999999999991e-102

   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 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.003 \lor \neg \left(y \leq 1.28 \cdot 10^{-102}\right):\\ \;\;\;\;\left(z + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11.5 \lor \neg \left(x \leq 2.15 \cdot 10^{-112}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -11.5) (not (<= x 2.15e-112))) (fma y x x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -11.5) || !(x <= 2.15e-112)) {
		tmp = fma(y, x, x);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -11.5) || !(x <= 2.15e-112))
		tmp = fma(y, x, x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -11.5], N[Not[LessEqual[x, 2.15e-112]], $MachinePrecision]], N[(y * x + x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -11.5 or 2.1499999999999999e-112 < 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 84.5

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

   5. Applied rewrites 84.5%

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

   if -11.5 < x < 2.1499999999999999e-112

   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 74.7

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

   5. Applied rewrites 74.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11.5 \lor \neg \left(x \leq 2.15 \cdot 10^{-112}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3900000 \lor \neg \left(x \leq 1.25 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3900000.0) (not (<= x 1.25e+55))) (* y x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3900000.0) || !(x <= 1.25e+55)) {
		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 <= (-3900000.0d0)) .or. (.not. (x <= 1.25d+55))) 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 <= -3900000.0) || !(x <= 1.25e+55)) {
		tmp = y * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3900000.0) or not (x <= 1.25e+55):
		tmp = y * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3900000.0) || !(x <= 1.25e+55))
		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 <= -3900000.0) || ~((x <= 1.25e+55)))
		tmp = y * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3900000.0], N[Not[LessEqual[x, 1.25e+55]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9e6 or 1.25000000000000011e55 < 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 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 42.7%

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

   if -3.9e6 < x < 1.25000000000000011e55

   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 67.9

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

   5. Applied rewrites 67.9%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3900000 \lor \neg \left(x \leq 1.25 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.1% 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 61.4

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

5. Applied rewrites 61.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))