Main:bigenough2 from A

Percentage Accurate: 100.0% → 98.9%

Time: 5.4s

Alternatives: 6

Speedup: 1.2×

• 20.7% 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: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+l+ N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. accelerator-lowering-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.75e+104)
   (fma y x x)
   (if (<= x 4.7e-34) (fma z y x) (fma y x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e+104) {
		tmp = fma(y, x, x);
	} else if (x <= 4.7e-34) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.75e+104)
		tmp = fma(y, x, x);
	elseif (x <= 4.7e-34)
		tmp = fma(z, y, x);
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.75e+104], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 4.7e-34], N[(z * y + x), $MachinePrecision], N[(y * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75000000000000008e104 or 4.70000000000000002e-34 < 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. accelerator-lowering-fma.f64 91.9

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

   5. Simplified 91.9%

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

   if -2.75000000000000008e104 < x < 4.70000000000000002e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 87.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 87.3

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

   7. Applied egg-rr 87.3%

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

Alternative 3: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-48}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-78}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e-48) (fma y x x) (if (<= x 6.5e-78) (* z y) (fma y x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e-48) {
		tmp = fma(y, x, x);
	} else if (x <= 6.5e-78) {
		tmp = z * y;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e-48)
		tmp = fma(y, x, x);
	elseif (x <= 6.5e-78)
		tmp = Float64(z * y);
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e-48], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 6.5e-78], N[(z * y), $MachinePrecision], N[(y * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-48 or 6.5000000000000003e-78 < 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. accelerator-lowering-fma.f64 85.5

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

   5. Simplified 85.5%

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

   if -1.2e-48 < x < 6.5000000000000003e-78

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot z + 0} \]

      2. accelerator-lowering-fma.f64 74.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, 0\right)} \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 74.0

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

   7. Applied egg-rr 74.0%

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

Alternative 4: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-21}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.2e-21) (* z y) (if (<= y 7e-8) x (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e-21) {
		tmp = z * y;
	} else if (y <= 7e-8) {
		tmp = 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 (y <= (-8.2d-21)) then
        tmp = z * y
    else if (y <= 7d-8) then
        tmp = 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 (y <= -8.2e-21) {
		tmp = z * y;
	} else if (y <= 7e-8) {
		tmp = x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.2e-21:
		tmp = z * y
	elif y <= 7e-8:
		tmp = x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.2e-21)
		tmp = Float64(z * y);
	elseif (y <= 7e-8)
		tmp = x;
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.2e-21)
		tmp = z * y;
	elseif (y <= 7e-8)
		tmp = x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.2e-21], N[(z * y), $MachinePrecision], If[LessEqual[y, 7e-8], x, N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.19999999999999988e-21 or 7.00000000000000048e-8 < 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. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot z + 0} \]

      2. accelerator-lowering-fma.f64 50.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, 0\right)} \]

   5. Simplified 50.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 50.8

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

   7. Applied egg-rr 50.8%

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

   if -8.19999999999999988e-21 < y < 7.00000000000000048e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 78.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 6: 37.1% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

5. Simplified 39.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))