Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 69.8% → 99.8%

Time: 1.1s

Alternatives: 3

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Initial Program: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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]

Alternative 1: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+129}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x\_m, x\_m, y\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x\_m}^{3}}, -0.125, \frac{0.5}{x\_m}\right), y, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 5e+129)
   (pow (fma x_m x_m y) 0.5)
   (fma (fma (/ y (pow x_m 3.0)) -0.125 (/ 0.5 x_m)) y x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 5e+129) {
		tmp = pow(fma(x_m, x_m, y), 0.5);
	} else {
		tmp = fma(fma((y / pow(x_m, 3.0)), -0.125, (0.5 / x_m)), y, x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 5e+129)
		tmp = fma(x_m, x_m, y) ^ 0.5;
	else
		tmp = fma(fma(Float64(y / (x_m ^ 3.0)), -0.125, Float64(0.5 / x_m)), y, x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 5e+129], N[Power[N[(x$95$m * x$95$m + y), $MachinePrecision], 0.5], $MachinePrecision], N[(N[(N[(y / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * -0.125 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] * y + x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000003e129

   1. Initial program 81.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fma.f64 81.2

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(x, x, y\right)\right)}}^{0.5} \]

   4. Applied rewrites 81.2%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, x, y\right)\right)}^{0.5}} \]

   if 5.0000000000000003e129 < x

   1. Initial program 12.3%

      \[\sqrt{x \cdot x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}, \color{blue}{y}, x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{{x}^{3}} \cdot \frac{-1}{8} + \frac{1}{2} \cdot \frac{1}{x}, y, x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2} \cdot 1}{x}\right), y, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{x}\right), y, x\right) \]

      10. lower-/.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, -0.125, \frac{0.5}{x}\right), y, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, -0.125, \frac{0.5}{x}\right), y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{x\_m \cdot x\_m + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x\_m}^{3}}, -0.125, \frac{0.5}{x\_m}\right), y, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 5e+129)
   (sqrt (+ (* x_m x_m) y))
   (fma (fma (/ y (pow x_m 3.0)) -0.125 (/ 0.5 x_m)) y x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 5e+129) {
		tmp = sqrt(((x_m * x_m) + y));
	} else {
		tmp = fma(fma((y / pow(x_m, 3.0)), -0.125, (0.5 / x_m)), y, x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 5e+129)
		tmp = sqrt(Float64(Float64(x_m * x_m) + y));
	else
		tmp = fma(fma(Float64(y / (x_m ^ 3.0)), -0.125, Float64(0.5 / x_m)), y, x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 5e+129], N[Sqrt[N[(N[(x$95$m * x$95$m), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], N[(N[(N[(y / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * -0.125 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] * y + x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000003e129

   1. Initial program 81.2%

      \[\sqrt{x \cdot x + y} \]
   2. Add Preprocessing

   if 5.0000000000000003e129 < x

   1. Initial program 12.3%

      \[\sqrt{x \cdot x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}, \color{blue}{y}, x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{{x}^{3}} \cdot \frac{-1}{8} + \frac{1}{2} \cdot \frac{1}{x}, y, x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2} \cdot 1}{x}\right), y, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{x}\right), y, x\right) \]

      10. lower-/.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, -0.125, \frac{0.5}{x}\right), y, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, -0.125, \frac{0.5}{x}\right), y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.0% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x\_m}^{3}}, -0.125, \frac{0.5}{x\_m}\right), y, x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (fma (fma (/ y (pow x_m 3.0)) -0.125 (/ 0.5 x_m)) y x_m))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return fma(fma((y / pow(x_m, 3.0)), -0.125, (0.5 / x_m)), y, x_m);
}

Julia

x_m = abs(x)
function code(x_m, y)
	return fma(fma(Float64(y / (x_m ^ 3.0)), -0.125, Float64(0.5 / x_m)), y, x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(N[(N[(y / N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * -0.125 + N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] * y + x$95$m), $MachinePrecision]

Derivation

1. Initial program 67.2%

   \[\sqrt{x \cdot x + y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + y \cdot \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto y \cdot \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) + \color{blue}{x} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right) \cdot y + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{y}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}, \color{blue}{y}, x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{y}{{x}^{3}} \cdot \frac{-1}{8} + \frac{1}{2} \cdot \frac{1}{x}, y, x\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{1}{2} \cdot \frac{1}{x}\right), y, x\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2} \cdot 1}{x}\right), y, x\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, \frac{-1}{8}, \frac{\frac{1}{2}}{x}\right), y, x\right) \]

   10. lower-/.f64 33.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, -0.125, \frac{0.5}{x}\right), y, x\right) \]

5. Applied rewrites 33.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{{x}^{3}}, -0.125, \frac{0.5}{x}\right), y, x\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1509769801047259300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ (* 1/2 (/ y x)) x)) (if (< x 5582399551122541000000000000000000000000000000000000000000) (sqrt (+ (* x x) y)) (+ (* 1/2 (/ y x)) x))))

  (sqrt (+ (* x x) y)))