bug366, discussion (missed optimization)

Percentage Accurate: 53.5% → 99.7%

Time: 3.2s

Alternatives: 2

Speedup: 24.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (- (* a a) (* b b))))

C

double code(double a, double b) {
	return sqrt(((a * a) - (b * b)));
}

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(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(((a * a) - (b * b)))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(((a * a) - (b * b)));
}

Python

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

Julia

function code(a, b)
	return sqrt(Float64(Float64(a * a) - Float64(b * b)))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(((a * a) - (b * b)));
end

Wolfram

code[a_, b_] := N[Sqrt[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (sqrt (- (* a a) (* b b))))

C

double code(double a, double b) {
	return sqrt(((a * a) - (b * b)));
}

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(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(((a * a) - (b * b)))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(((a * a) - (b * b)));
}

Python

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

Julia

function code(a, b)
	return sqrt(Float64(Float64(a * a) - Float64(b * b)))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(((a * a) - (b * b)));
end

Wolfram

code[a_, b_] := N[Sqrt[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \mathsf{fma}\left(\frac{-0.5}{a\_m} \cdot b, b, a\_m\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (fma (* (/ -0.5 a_m) b) b a_m))

C

a_m = fabs(a);
double code(double a_m, double b) {
	return fma(((-0.5 / a_m) * b), b, a_m);
}

Julia

a_m = abs(a)
function code(a_m, b)
	return fma(Float64(Float64(-0.5 / a_m) * b), b, a_m)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(N[(N[(-0.5 / a$95$m), $MachinePrecision] * b), $MachinePrecision] * b + a$95$m), $MachinePrecision]

Derivation

1. Initial program 49.9%

   \[\sqrt{a \cdot a - b \cdot b} \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{a + \frac{-1}{2} \cdot \frac{{b}^{2}}{a}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{b}^{2}}{a} + a} \]

   2. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot {b}^{2}}{a}} + a \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a} \cdot {b}^{2}} + a \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot 1}}{a} \cdot {b}^{2} + a \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{a}\right)} \cdot {b}^{2} + a \]

   6. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{a}\right) \cdot {b}^{2} + a \]

   7. unpow2 N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{a}\right) \cdot \color{blue}{\left(b \cdot b\right)} + a \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{a}\right) \cdot b\right) \cdot b} + a \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{a}\right) \cdot b, b, a\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{a}\right) \cdot b}, b, a\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{a}\right) \cdot b, b, a\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2} \cdot 1}{a}} \cdot b, b, a\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2}}}{a} \cdot b, b, a\right) \]

   14. lower-/.f64 42.7

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-0.5}{a}} \cdot b, b, a\right) \]

5. Applied rewrites 42.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{a} \cdot b, b, a\right)} \]

6. Final simplification 42.7%

   \[\leadsto \mathsf{fma}\left(\frac{-0.5}{a} \cdot b, b, a\right) \]
7. Add Preprocessing

Alternative 2: 99.2% accurate, 24.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a\_m \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 a_m)

C

a_m = fabs(a);
double code(double a_m, double b) {
	return a_m;
}

Fortran

a_m =     private
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(a_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = a_m
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return a_m;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return a_m

Julia

a_m = abs(a)
function code(a_m, b)
	return a_m
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b)
	tmp = a_m;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := a$95$m

Derivation

1. Initial program 49.9%

   \[\sqrt{a \cdot a - b \cdot b} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \sqrt{\color{blue}{a \cdot a - b \cdot b}} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{a \cdot a} - b \cdot b} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{a \cdot a - \color{blue}{b \cdot b}} \]

   4. difference-of-squares N/A

      \[\leadsto \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}} \]

   7. lower--.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(a - b\right)} \cdot \left(a + b\right)} \]

   8. +-commutative N/A

      \[\leadsto \sqrt{\left(a - b\right) \cdot \color{blue}{\left(b + a\right)}} \]

   9. lower-+.f64 50.3

      \[\leadsto \sqrt{\left(a - b\right) \cdot \color{blue}{\left(b + a\right)}} \]

4. Applied rewrites 50.3%

   \[\leadsto \sqrt{\color{blue}{\left(a - b\right) \cdot \left(b + a\right)}} \]

5. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot \frac{b + -1 \cdot b}{a}\right)} \]
6. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto a \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{b + -1 \cdot b}{a}\right)} \]

   2. metadata-eval N/A

      \[\leadsto a \cdot \left(1 - \color{blue}{\frac{-1}{2}} \cdot \frac{b + -1 \cdot b}{a}\right) \]

   3. *-commutative N/A

      \[\leadsto a \cdot \left(1 - \color{blue}{\frac{b + -1 \cdot b}{a} \cdot \frac{-1}{2}}\right) \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto a \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{b + -1 \cdot b}{a}\right)\right) \cdot \frac{-1}{2}\right)} \]

   5. distribute-frac-neg N/A

      \[\leadsto a \cdot \left(1 + \color{blue}{\frac{\mathsf{neg}\left(\left(b + -1 \cdot b\right)\right)}{a}} \cdot \frac{-1}{2}\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto a \cdot \left(1 + \frac{\mathsf{neg}\left(\color{blue}{\left(-1 + 1\right) \cdot b}\right)}{a} \cdot \frac{-1}{2}\right) \]

   7. metadata-eval N/A

      \[\leadsto a \cdot \left(1 + \frac{\mathsf{neg}\left(\color{blue}{0} \cdot b\right)}{a} \cdot \frac{-1}{2}\right) \]

   8. distribute-lft-neg-in N/A

      \[\leadsto a \cdot \left(1 + \frac{\color{blue}{\left(\mathsf{neg}\left(0\right)\right) \cdot b}}{a} \cdot \frac{-1}{2}\right) \]

   9. metadata-eval N/A

      \[\leadsto a \cdot \left(1 + \frac{\color{blue}{0} \cdot b}{a} \cdot \frac{-1}{2}\right) \]

   10. metadata-eval N/A

       \[\leadsto a \cdot \left(1 + \frac{\color{blue}{\left(-1 + 1\right)} \cdot b}{a} \cdot \frac{-1}{2}\right) \]

   11. distribute-rgt1-in N/A

       \[\leadsto a \cdot \left(1 + \frac{\color{blue}{b + -1 \cdot b}}{a} \cdot \frac{-1}{2}\right) \]

   12. div-add N/A

       \[\leadsto a \cdot \left(1 + \color{blue}{\left(\frac{b}{a} + \frac{-1 \cdot b}{a}\right)} \cdot \frac{-1}{2}\right) \]

   13. associate-*r/ N/A

       \[\leadsto a \cdot \left(1 + \left(\frac{b}{a} + \color{blue}{-1 \cdot \frac{b}{a}}\right) \cdot \frac{-1}{2}\right) \]

   14. distribute-rgt1-in N/A

       \[\leadsto a \cdot \left(1 + \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{b}{a}\right)} \cdot \frac{-1}{2}\right) \]

   15. metadata-eval N/A

       \[\leadsto a \cdot \left(1 + \left(\color{blue}{0} \cdot \frac{b}{a}\right) \cdot \frac{-1}{2}\right) \]

   16. mul0-lft N/A

       \[\leadsto a \cdot \left(1 + \color{blue}{0} \cdot \frac{-1}{2}\right) \]

   17. metadata-eval N/A

       \[\leadsto a \cdot \left(1 + \color{blue}{0}\right) \]

   18. metadata-eval N/A

       \[\leadsto a \cdot \color{blue}{1} \]

7. Applied rewrites 42.2%

   \[\leadsto \color{blue}{a} \]
8. Add Preprocessing

Developer Target 1: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|a\right| + \left|b\right|} \cdot \sqrt{\left|a\right| - \left|b\right|} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (sqrt (+ (fabs a) (fabs b))) (sqrt (- (fabs a) (fabs b)))))

C

double code(double a, double b) {
	return sqrt((fabs(a) + fabs(b))) * sqrt((fabs(a) - fabs(b)));
}

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(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt((abs(a) + abs(b))) * sqrt((abs(a) - abs(b)))
end function

Java

public static double code(double a, double b) {
	return Math.sqrt((Math.abs(a) + Math.abs(b))) * Math.sqrt((Math.abs(a) - Math.abs(b)));
}

Python

def code(a, b):
	return math.sqrt((math.fabs(a) + math.fabs(b))) * math.sqrt((math.fabs(a) - math.fabs(b)))

Julia

function code(a, b)
	return Float64(sqrt(Float64(abs(a) + abs(b))) * sqrt(Float64(abs(a) - abs(b))))
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt((abs(a) + abs(b))) * sqrt((abs(a) - abs(b)));
end

Wolfram

code[a_, b_] := N[(N[Sqrt[N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[a], $MachinePrecision] - N[Abs[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024356 
(FPCore (a b)
  :name "bug366, discussion (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((fa (fabs a)) (fb (fabs b))) (* (sqrt (+ fa fb)) (sqrt (- fa fb)))))

  (sqrt (- (* a a) (* b b))))