A quarter-circle in the lower-left quadrant

Percentage Accurate: 100.0% → 100.0%
Time: 1.8s
Alternatives: 2
Speedup: 2.3×

Specification

?
\[\mathsf{max}\left(\left({y}^{2} + {x}^{2}\right) - 0.5, \mathsf{max}\left(x, y\right)\right) \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fmax (- (+ (pow y 2.0) (pow x 2.0)) 0.5) (fmax x y)))
double code(double x, double y) {
	return fmax(((pow(y, 2.0) + pow(x, 2.0)) - 0.5), fmax(x, y));
}

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = fmax((((y ** 2.0d0) + (x ** 2.0d0)) - 0.5d0), fmax(x, y))
end function

public static double code(double x, double y) {
	return fmax(((Math.pow(y, 2.0) + Math.pow(x, 2.0)) - 0.5), fmax(x, y));
}

def code(x, y):
	return fmax(((math.pow(y, 2.0) + math.pow(x, 2.0)) - 0.5), fmax(x, y))

function code(x, y)
	return fmax(Float64(Float64((y ^ 2.0) + (x ^ 2.0)) - 0.5), fmax(x, y))
end

function tmp = code(x, y)
	tmp = max((((y ^ 2.0) + (x ^ 2.0)) - 0.5), max(x, y));
end

code[x_, y_] := N[Max[N[(N[(N[Power[y, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[Max[x, y], $MachinePrecision]], $MachinePrecision]

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp_1 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp_2 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp = IF ((((y ^ (2)) + (x ^ (2))) - (5e-1)) > tmp_1) THEN (((y ^ (2)) + (x ^ (2))) - (5e-1)) ELSE tmp_2 ENDIF IN
	tmp
END code
\mathsf{max}\left(\left({y}^{2} + {x}^{2}\right) - 0.5, \mathsf{max}\left(x, y\right)\right)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\mathsf{max}\left(\left({y}^{2} + {x}^{2}\right) - 0.5, \mathsf{max}\left(x, y\right)\right) \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fmax (- (+ (pow y 2.0) (pow x 2.0)) 0.5) (fmax x y)))
double code(double x, double y) {
	return fmax(((pow(y, 2.0) + pow(x, 2.0)) - 0.5), fmax(x, y));
}

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = fmax((((y ** 2.0d0) + (x ** 2.0d0)) - 0.5d0), fmax(x, y))
end function

public static double code(double x, double y) {
	return fmax(((Math.pow(y, 2.0) + Math.pow(x, 2.0)) - 0.5), fmax(x, y));
}

def code(x, y):
	return fmax(((math.pow(y, 2.0) + math.pow(x, 2.0)) - 0.5), fmax(x, y))

function code(x, y)
	return fmax(Float64(Float64((y ^ 2.0) + (x ^ 2.0)) - 0.5), fmax(x, y))
end

function tmp = code(x, y)
	tmp = max((((y ^ 2.0) + (x ^ 2.0)) - 0.5), max(x, y));
end

code[x_, y_] := N[Max[N[(N[(N[Power[y, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[Max[x, y], $MachinePrecision]], $MachinePrecision]

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp_1 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp_2 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp = IF ((((y ^ (2)) + (x ^ (2))) - (5e-1)) > tmp_1) THEN (((y ^ (2)) + (x ^ (2))) - (5e-1)) ELSE tmp_2 ENDIF IN
	tmp
END code
\mathsf{max}\left(\left({y}^{2} + {x}^{2}\right) - 0.5, \mathsf{max}\left(x, y\right)\right)

Alternative 1: 100.0% accurate, 2.3× speedup?

\[\mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, -0.5\right)\right), \mathsf{max}\left(x, y\right)\right) \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fmax (* 1.0 (fma x x (fma y y -0.5))) (fmax x y)))
double code(double x, double y) {
	return fmax((1.0 * fma(x, x, fma(y, y, -0.5))), fmax(x, y));
}

function code(x, y)
	return fmax(Float64(1.0 * fma(x, x, fma(y, y, -0.5))), fmax(x, y))
end

code[x_, y_] := N[Max[N[(1.0 * N[(x * x + N[(y * y + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Max[x, y], $MachinePrecision]], $MachinePrecision]

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp_1 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp_2 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp = IF (((1) * ((x * x) + ((y * y) + (-5e-1)))) > tmp_1) THEN ((1) * ((x * x) + ((y * y) + (-5e-1)))) ELSE tmp_2 ENDIF IN
	tmp
END code
\mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, -0.5\right)\right), \mathsf{max}\left(x, y\right)\right)
Derivation

  1. Initial program 100.0%

    \[\mathsf{max}\left(\left({y}^{2} + {x}^{2}\right) - 0.5, \mathsf{max}\left(x, y\right)\right) \]
  2. Step-by-step derivation

    1. Applied rewrites100.0%

      \[\leadsto \mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, -0.5\right)\right), \mathsf{max}\left(x, y\right)\right) \]
    2. Add Preprocessing

    Alternative 2: 67.7% accurate, 3.1× speedup?

    \[\mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, -0.5\right), \mathsf{max}\left(x, y\right)\right) \]
    (FPCore (x y)
  :precision binary64
  :pre TRUE
  (fmax (* 1.0 (fma x x -0.5)) (fmax x y)))
    double code(double x, double y) {
	return fmax((1.0 * fma(x, x, -0.5)), fmax(x, y));
}

    function code(x, y)
	return fmax(Float64(1.0 * fma(x, x, -0.5)), fmax(x, y))
end

    code[x_, y_] := N[Max[N[(1.0 * N[(x * x + -0.5), $MachinePrecision]), $MachinePrecision], N[Max[x, y], $MachinePrecision]], $MachinePrecision]

    f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp_1 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp_2 = IF (x > y) THEN x ELSE y ENDIF IN
	LET tmp = IF (((1) * ((x * x) + (-5e-1))) > tmp_1) THEN ((1) * ((x * x) + (-5e-1))) ELSE tmp_2 ENDIF IN
	tmp
END code
    \mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, -0.5\right), \mathsf{max}\left(x, y\right)\right)
    Derivation

    1. Initial program 100.0%

      \[\mathsf{max}\left(\left({y}^{2} + {x}^{2}\right) - 0.5, \mathsf{max}\left(x, y\right)\right) \]
    2. Step-by-step derivation

      1. Applied rewrites100.0%

        \[\leadsto \mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, -0.5\right)\right), \mathsf{max}\left(x, y\right)\right) \]

      2. Taylor expanded in y around 0

        \[\leadsto \mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, \frac{-1}{2}\right), \mathsf{max}\left(x, y\right)\right) \]
      3. Step-by-step derivation

        1. Applied rewrites67.7%

          \[\leadsto \mathsf{max}\left(1 \cdot \mathsf{fma}\left(x, x, -0.5\right), \mathsf{max}\left(x, y\right)\right) \]
        2. Add Preprocessing

        Reproduce

        ?
        herbie shell --seed 2026070 
(FPCore (x y)
  :name "A quarter-circle in the lower-left quadrant"
  :precision binary64
  (fmax (- (+ (pow y 2.0) (pow x 2.0)) 0.5) (fmax x y)))