Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 7

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

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

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. +-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. Add Preprocessing

Alternative 2: 61.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+210}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-28}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+87}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+210)
   (* x y)
   (if (<= y -1.4e-28)
     (* z y)
     (if (<= y 3.4e-43) x (if (<= y 1.05e+87) (* z y) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+210) {
		tmp = x * y;
	} else if (y <= -1.4e-28) {
		tmp = z * y;
	} else if (y <= 3.4e-43) {
		tmp = x;
	} else if (y <= 1.05e+87) {
		tmp = z * y;
	} else {
		tmp = x * 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 <= (-1.8d+210)) then
        tmp = x * y
    else if (y <= (-1.4d-28)) then
        tmp = z * y
    else if (y <= 3.4d-43) then
        tmp = x
    else if (y <= 1.05d+87) then
        tmp = z * y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+210) {
		tmp = x * y;
	} else if (y <= -1.4e-28) {
		tmp = z * y;
	} else if (y <= 3.4e-43) {
		tmp = x;
	} else if (y <= 1.05e+87) {
		tmp = z * y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.8e+210:
		tmp = x * y
	elif y <= -1.4e-28:
		tmp = z * y
	elif y <= 3.4e-43:
		tmp = x
	elif y <= 1.05e+87:
		tmp = z * y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+210)
		tmp = Float64(x * y);
	elseif (y <= -1.4e-28)
		tmp = Float64(z * y);
	elseif (y <= 3.4e-43)
		tmp = x;
	elseif (y <= 1.05e+87)
		tmp = Float64(z * y);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.8e+210)
		tmp = x * y;
	elseif (y <= -1.4e-28)
		tmp = z * y;
	elseif (y <= 3.4e-43)
		tmp = x;
	elseif (y <= 1.05e+87)
		tmp = z * y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+210], N[(x * y), $MachinePrecision], If[LessEqual[y, -1.4e-28], N[(z * y), $MachinePrecision], If[LessEqual[y, 3.4e-43], x, If[LessEqual[y, 1.05e+87], N[(z * y), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8000000000000001e210 or 1.05e87 < y

   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 63.4

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

   5. Simplified 63.4%

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

   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. *-lowering-*.f64 63.4

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

   8. Simplified 63.4%

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

   if -1.8000000000000001e210 < y < -1.3999999999999999e-28 or 3.4000000000000001e-43 < y < 1.05e87

   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. *-lowering-*.f64 61.4

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

   5. Simplified 61.4%

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

   if -1.3999999999999999e-28 < y < 3.4000000000000001e-43

   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 77.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+210}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-28}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+87}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ \mathbf{if}\;y \leq -52000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ x z))))
   (if (<= y -52000000.0) t_0 (if (<= y 1.3e-16) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double tmp;
	if (y <= -52000000.0) {
		tmp = t_0;
	} else if (y <= 1.3e-16) {
		tmp = fma(z, y, 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 <= -52000000.0)
		tmp = t_0;
	elseif (y <= 1.3e-16)
		tmp = fma(z, y, 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, -52000000.0], t$95$0, If[LessEqual[y, 1.3e-16], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e7 or 1.2999999999999999e-16 < 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. *-lowering-*.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. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

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

   if -5.2e7 < y < 1.2999999999999999e-16

   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. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -52000000:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-152}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e+115)
   (fma y x x)
   (if (<= x 3.2e-152) (fma z y x) (fma y x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e+115) {
		tmp = fma(y, x, x);
	} else if (x <= 3.2e-152) {
		tmp = fma(z, y, x);
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.2e+115)
		tmp = fma(y, x, x);
	elseif (x <= 3.2e-152)
		tmp = fma(z, y, x);
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.2e+115], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 3.2e-152], N[(z * y + x), $MachinePrecision], N[(y * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000007e115 or 3.20000000000000013e-152 < 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 86.9

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

   5. Simplified 86.9%

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

   if -4.20000000000000007e115 < x < 3.20000000000000013e-152

   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. Taylor expanded in x around 0

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

   7. Simplified 88.3%

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

Alternative 5: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-152}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.8e-26) (fma y x x) (if (<= x 3e-152) (* z y) (fma y x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8e-26) {
		tmp = fma(y, x, x);
	} else if (x <= 3e-152) {
		tmp = z * y;
	} else {
		tmp = fma(y, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.8e-26)
		tmp = fma(y, x, x);
	elseif (x <= 3e-152)
		tmp = Float64(z * y);
	else
		tmp = fma(y, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.8e-26], N[(y * x + x), $MachinePrecision], If[LessEqual[x, 3e-152], N[(z * y), $MachinePrecision], N[(y * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e-26 or 3e-152 < 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 82.9

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

   5. Simplified 82.9%

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

   if -2.8000000000000001e-26 < x < 3e-152

   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. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-152}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.8e-15) (* x y) (if (<= y 6.7e-15) x (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.8e-15) {
		tmp = x * y;
	} else if (y <= 6.7e-15) {
		tmp = x;
	} else {
		tmp = x * 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 <= (-2.8d-15)) then
        tmp = x * y
    else if (y <= 6.7d-15) then
        tmp = x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.8e-15) {
		tmp = x * y;
	} else if (y <= 6.7e-15) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.8e-15:
		tmp = x * y
	elif y <= 6.7e-15:
		tmp = x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.8e-15)
		tmp = Float64(x * y);
	elseif (y <= 6.7e-15)
		tmp = x;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.8e-15)
		tmp = x * y;
	elseif (y <= 6.7e-15)
		tmp = x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.8e-15], N[(x * y), $MachinePrecision], If[LessEqual[y, 6.7e-15], x, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000014e-15 or 6.70000000000000001e-15 < y

   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 54.3

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

   5. Simplified 54.3%

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

   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. *-lowering-*.f64 53.8

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

   8. Simplified 53.8%

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

   if -2.80000000000000014e-15 < y < 6.70000000000000001e-15

   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 74.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.4% 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 36.0%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))