Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 6

Speedup: 1.2×

• 21.2% 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

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 100.0

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.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. Add Preprocessing

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + z\right) \cdot y\\ \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x z) y)))
   (if (<= y -75000000.0) t_0 (if (<= y 1.0) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + z) * y;
	double tmp;
	if (y <= -75000000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + z) * y)
	tmp = 0.0
	if (y <= -75000000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -75000000.0], t$95$0, If[LessEqual[y, 1.0], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e7 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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 98.5

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

   5. Simplified 98.5%

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

   if -7.5e7 < 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. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -75000000:\\ \;\;\;\;\left(x + z\right) \cdot y\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-128}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e-128)
   (fma z y x)
   (if (<= z 4.6e-115) (fma y x x) (fma z y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e-128) {
		tmp = fma(z, y, x);
	} else if (z <= 4.6e-115) {
		tmp = fma(y, x, x);
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e-128)
		tmp = fma(z, y, x);
	elseif (z <= 4.6e-115)
		tmp = fma(y, x, x);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e-128], N[(z * y + x), $MachinePrecision], If[LessEqual[z, 4.6e-115], N[(y * x + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000002e-128 or 4.59999999999999969e-115 < z

   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. lower-*.f64 91.6

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

   5. Simplified 91.6%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 91.6

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

   7. Applied egg-rr 91.6%

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

   if -3.8000000000000002e-128 < z < 4.59999999999999969e-115

   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 97.1

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

   5. Simplified 97.1%

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

Alternative 4: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+115}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 31500000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.6e+115)
   (* z y)
   (if (<= z 31500000000000.0) (fma y x x) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.6e+115) {
		tmp = z * y;
	} else if (z <= 31500000000000.0) {
		tmp = fma(y, x, x);
	} else {
		tmp = z * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.6e+115)
		tmp = Float64(z * y);
	elseif (z <= 31500000000000.0)
		tmp = fma(y, x, x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.6e+115], N[(z * y), $MachinePrecision], If[LessEqual[z, 31500000000000.0], N[(y * x + x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6000000000000001e115 or 3.15e13 < z

   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. lower-*.f64 76.7

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

   5. Simplified 76.7%

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

   if -7.6000000000000001e115 < z < 3.15e13

   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 79.1

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

   5. Simplified 79.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+115}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 31500000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-128}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.1e-128) (* z y) (if (<= z 4.6e-115) (* x y) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e-128) {
		tmp = z * y;
	} else if (z <= 4.6e-115) {
		tmp = x * y;
	} 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 (z <= (-3.1d-128)) then
        tmp = z * y
    else if (z <= 4.6d-115) then
        tmp = x * y
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.1e-128) {
		tmp = z * y;
	} else if (z <= 4.6e-115) {
		tmp = x * y;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.1e-128:
		tmp = z * y
	elif z <= 4.6e-115:
		tmp = x * y
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.1e-128)
		tmp = Float64(z * y);
	elseif (z <= 4.6e-115)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.1e-128)
		tmp = z * y;
	elseif (z <= 4.6e-115)
		tmp = x * y;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.1e-128], N[(z * y), $MachinePrecision], If[LessEqual[z, 4.6e-115], N[(x * y), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000003e-128 or 4.59999999999999969e-115 < z

   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. lower-*.f64 60.2

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

   5. Simplified 60.2%

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

   if -3.10000000000000003e-128 < z < 4.59999999999999969e-115

   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 97.1

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

   5. Simplified 97.1%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.4

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

   8. Simplified 46.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-128}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * y), $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 57.5

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

5. Simplified 57.5%

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

6. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 22.2

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

8. Simplified 22.2%

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

9. Final simplification 22.2%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))