Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 6

Speedup: 1.2×

• 20.6% 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

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

FPCore

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

C

double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (y * z) + x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y * (x + z)
    if (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = (y * z) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	t_0 = y * (x + z)
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = (y * z) + x
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x + z);
	tmp = 0.0;
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = (y * z) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

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 98.3

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

   5. Applied rewrites 98.3%

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

   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 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.5%

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

Alternative 3: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ \mathbf{if}\;y \leq -3400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ x z))))
   (if (<= y -3400.0) t_0 (if (<= y 2.4e-126) (fma y x x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double tmp;
	if (y <= -3400.0) {
		tmp = t_0;
	} else if (y <= 2.4e-126) {
		tmp = fma(y, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x + z))
	tmp = 0.0
	if (y <= -3400.0)
		tmp = t_0;
	elseif (y <= 2.4e-126)
		tmp = fma(y, x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3400.0], t$95$0, If[LessEqual[y, 2.4e-126], N[(y * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3400 or 2.40000000000000007e-126 < 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 93.9

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

   5. Applied rewrites 93.9%

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

   if -3400 < y < 2.40000000000000007e-126

   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 79.9

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

   5. Applied rewrites 79.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3400:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-122}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e-114) (fma y x x) (if (<= x 2.3e-122) (* y z) (fma y x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e-114) {
		tmp = fma(y, x, x);
	} else if (x <= 2.3e-122) {
		tmp = y * z;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e-114)
		tmp = fma(y, x, x);
	elseif (x <= 2.3e-122)
		tmp = Float64(y * z);
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e-114], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 2.3e-122], N[(y * z), $MachinePrecision], N[(y * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-114 or 2.30000000000000007e-122 < 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 79.9

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

   5. Applied rewrites 79.9%

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

   if -1.15e-114 < x < 2.30000000000000007e-122

   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 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-122}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+121}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e-65) (* y x) (if (<= x 3.5e+121) (* y z) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-65) {
		tmp = y * x;
	} else if (x <= 3.5e+121) {
		tmp = y * z;
	} else {
		tmp = y * 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 (x <= (-1.45d-65)) then
        tmp = y * x
    else if (x <= 3.5d+121) then
        tmp = y * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e-65) {
		tmp = y * x;
	} else if (x <= 3.5e+121) {
		tmp = y * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45e-65:
		tmp = y * x
	elif x <= 3.5e+121:
		tmp = y * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e-65)
		tmp = Float64(y * x);
	elseif (x <= 3.5e+121)
		tmp = Float64(y * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e-65)
		tmp = y * x;
	elseif (x <= 3.5e+121)
		tmp = y * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e-65], N[(y * x), $MachinePrecision], If[LessEqual[x, 3.5e+121], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4499999999999999e-65 or 3.5e121 < 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 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 43.4%

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

   if -1.4499999999999999e-65 < x < 3.5e121

   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 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+121}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.0% 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 63.0

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

5. Applied rewrites 63.0%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 29.1%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))