Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 68.9% → 99.4%

Time: 1.7s

Alternatives: 5

Speedup: 0.6×

Specification

Math

\[\sqrt{x \cdot x + y} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (sqrt (+ (* x x) y)))

C

double code(double x, double y) {
	return sqrt(((x * x) + y));
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(((x * x) + y))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(((x * x) + y));
}

Python

def code(x, y):
	return math.sqrt(((x * x) + y))

Julia

function code(x, y)
	return sqrt(Float64(Float64(x * x) + y))
end

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	sqrt(((x * x) + y))
END code

Initial Program: 68.9% accurate, 1.0× speedup

Math

\[\sqrt{x \cdot x + y} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (sqrt (+ (* x x) y)))

C

double code(double x, double y) {
	return sqrt(((x * x) + y));
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(((x * x) + y))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(((x * x) + y));
}

Python

def code(x, y):
	return math.sqrt(((x * x) + y))

Julia

function code(x, y)
	return sqrt(Float64(Float64(x * x) + y))
end

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	sqrt(((x * x) + y))
END code

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \cdot \left|x\right| \leq 10^{+200}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left|x\right|, \left|x\right|, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (* (fabs x) (fabs x)) 1e+200)
  (sqrt (fma (fabs x) (fabs x) y))
  (* (fabs x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((fabs(x) * fabs(x)) <= 1e+200) {
		tmp = sqrt(fma(fabs(x), fabs(x), y));
	} else {
		tmp = fabs(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(x) * abs(x)) <= 1e+200)
		tmp = sqrt(fma(abs(x), abs(x), y));
	else
		tmp = Float64(abs(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], 1e+200], N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * 1.0), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((abs(x)) * (abs(x))) <= (99999999999999996973312221251036165947450327545502362648241750950346848435554075534196338404706251868027512415973882408182135734368278484639385041047239877871023591066789981811181813306167128854888448)) THEN (sqrt((((abs(x)) * (abs(x))) + y))) ELSE ((abs(x)) * (1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999997e199

   1. Initial program 68.9%

      \[\sqrt{x \cdot x + y} \]
   2. Step-by-step derivation

   3. Applied rewrites 68.9%

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

   if 9.9999999999999997e199 < (*.f64 x x)

   1. Initial program 68.9%

      \[\sqrt{x \cdot x + y} \]

   2. Taylor expanded in x around inf

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 34.4%

      \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{y}{{x}^{2}}\right) \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 34.4%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \cdot \left|x\right| \leq 2 \cdot 10^{-99}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot 1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (* (fabs x) (fabs x)) 2e-99) (sqrt y) (* (fabs x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((fabs(x) * fabs(x)) <= 2e-99) {
		tmp = sqrt(y);
	} else {
		tmp = fabs(x) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((abs(x) * abs(x)) <= 2d-99) then
        tmp = sqrt(y)
    else
        tmp = abs(x) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.abs(x) * Math.abs(x)) <= 2e-99) {
		tmp = Math.sqrt(y);
	} else {
		tmp = Math.abs(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.fabs(x) * math.fabs(x)) <= 2e-99:
		tmp = math.sqrt(y)
	else:
		tmp = math.fabs(x) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(abs(x) * abs(x)) <= 2e-99)
		tmp = sqrt(y);
	else
		tmp = Float64(abs(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((abs(x) * abs(x)) <= 2e-99)
		tmp = sqrt(y);
	else
		tmp = abs(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], 2e-99], N[Sqrt[y], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * 1.0), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((abs(x)) * (abs(x))) <= (20000000000000000399837996052057672392955215770683188403652060118731913985110869352353525772265859791654921496218237015970565410794993080445368720839225272167125662825574358854498578849381613317832611860008691572046029005015889997371182867751115974641606953809969127178192138671875e-379)) THEN (sqrt(y)) ELSE ((abs(x)) * (1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e-99

   1. Initial program 68.9%

      \[\sqrt{x \cdot x + y} \]

   2. Taylor expanded in x around 0

      \[\leadsto \sqrt{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 34.6%

      \[\leadsto \sqrt{y} \]

   if 2e-99 < (*.f64 x x)

   1. Initial program 68.9%

      \[\sqrt{x \cdot x + y} \]

   2. Taylor expanded in x around inf

      \[\leadsto x \cdot \left(1 + \frac{1}{2} \cdot \frac{y}{{x}^{2}}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 34.4%

      \[\leadsto x \cdot \left(1 + 0.5 \cdot \frac{y}{{x}^{2}}\right) \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 34.4%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 34.7% accurate, 2.9× speedup

Math

\[\sqrt{y} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (sqrt y))

C

double code(double x, double y) {
	return sqrt(y);
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(y)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(y);
}

Python

def code(x, y):
	return math.sqrt(y)

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[Sqrt[y], $MachinePrecision]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	sqrt(y)
END code

Derivation

1. Initial program 68.9%

   \[\sqrt{x \cdot x + y} \]

2. Taylor expanded in x around 0

   \[\leadsto \sqrt{y} \]
3. Step-by-step derivation

4. Applied rewrites 34.6%

   \[\leadsto \sqrt{y} \]
5. Add Preprocessing

Alternative 4: 34.6% accurate, 4.3× speedup

Math

\[-0 \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (- 0.0))

C

double code(double x, double y) {
	return -0.0;
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -0.0d0
end function

Java

public static double code(double x, double y) {
	return -0.0;
}

Python

def code(x, y):
	return -0.0

Julia

function code(x, y)
	return Float64(-0.0)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.0;
end

Wolfram

code[x_, y_] := (-0.0)

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	- (0)
END code

Derivation

1. Initial program 68.9%

   \[\sqrt{x \cdot x + y} \]

2. Taylor expanded in x around -inf

   \[\leadsto -1 \cdot x \]
3. Step-by-step derivation

4. Applied rewrites 34.7%

   \[\leadsto -1 \cdot x \]
5. Step-by-step derivation

6. Applied rewrites 34.7%

   \[\leadsto -x \]

7. Taylor expanded in undef-var around zero

   \[\leadsto -0 \]
8. Step-by-step derivation

9. Applied rewrites 2.6%

   \[\leadsto -0 \]
10. Add Preprocessing

Alternative 5: 2.6% accurate, 4.3× speedup

Math

\[-x \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (- x))

C

double code(double x, double y) {
	return -x;
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -x
end function

Java

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

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_, y_] := (-x)

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	- x
END code

Derivation

1. Initial program 68.9%

   \[\sqrt{x \cdot x + y} \]

2. Taylor expanded in x around -inf

   \[\leadsto -1 \cdot x \]
3. Step-by-step derivation

4. Applied rewrites 34.7%

   \[\leadsto -1 \cdot x \]
5. Step-by-step derivation

6. Applied rewrites 34.7%

   \[\leadsto -x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64
  (sqrt (+ (* x x) y)))