Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 6

Speedup: 1.2×

• 21.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x + z), $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. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y (+ x z)) (+ (* y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (x + z);
	} else {
		tmp = (y * 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 ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (x + z)
    else
        tmp = (y * z) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (x + z);
	} else {
		tmp = (y * z) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = y * (x + z)
	else:
		tmp = (y * z) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = Float64(Float64(y * z) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y * (x + z);
	else
		tmp = (y * z) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x + z\right)} \]
   4. 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 98.8

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

   5. Applied rewrites 98.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-28} \lor \neg \left(y \leq 0.0135\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e-28) (not (<= y 0.0135))) (* y (+ x z)) (fma y x x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-28) || !(y <= 0.0135)) {
		tmp = y * (x + z);
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e-28) || !(y <= 0.0135))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e-28], N[Not[LessEqual[y, 0.0135]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(y * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999996e-28 or 0.0134999999999999998 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x + z\right)} \]
   4. 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 97.4

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

   5. Applied rewrites 97.4%

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

   if -2.19999999999999996e-28 < y < 0.0134999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-28} \lor \neg \left(y \leq 0.0135\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-61} \lor \neg \left(x \leq 5.2 \cdot 10^{-124}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.1e-61) (not (<= x 5.2e-124))) (fma y x x) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.1e-61) || !(x <= 5.2e-124)) {
		tmp = fma(y, x, x);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.1e-61) || !(x <= 5.2e-124))
		tmp = fma(y, x, x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.1e-61], N[Not[LessEqual[x, 5.2e-124]], $MachinePrecision]], N[(y * x + x), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999999e-61 or 5.1999999999999999e-124 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -4.09999999999999999e-61 < x < 5.1999999999999999e-124

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-61} \lor \neg \left(x \leq 5.2 \cdot 10^{-124}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+22} \lor \neg \left(x \leq 1.05 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e+22) (not (<= x 1.05e+79))) (* y x) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e+22) || !(x <= 1.05e+79)) {
		tmp = y * x;
	} else {
		tmp = y * 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 <= (-3.7d+22)) .or. (.not. (x <= 1.05d+79))) then
        tmp = y * x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e+22) || !(x <= 1.05e+79)) {
		tmp = y * x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7e+22) or not (x <= 1.05e+79):
		tmp = y * x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e+22) || !(x <= 1.05e+79))
		tmp = Float64(y * x);
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.7e+22) || ~((x <= 1.05e+79)))
		tmp = y * x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e+22], N[Not[LessEqual[x, 1.05e+79]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6999999999999998e22 or 1.05000000000000004e79 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 51.0%

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

   if -3.6999999999999998e22 < x < 1.05000000000000004e79

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+22} \lor \neg \left(x \leq 1.05 \cdot 10^{+79}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.5% 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 z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 31.0%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))