Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2

Percentage Accurate: 99.9% → 100.0%
Time: 1.2s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\frac{x + y}{y + y} \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (+ x y) (+ y y)))
double code(double x, double y) {
	return (x + y) / (y + y);
}

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

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

def code(x, y):
	return (x + y) / (y + y)

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + y))
end

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

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(x + y) / (y + y)
END code
\frac{x + y}{y + y}

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 3 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: 99.9% accurate, 1.0× speedup?

\[\frac{x + y}{y + y} \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (+ x y) (+ y y)))
double code(double x, double y) {
	return (x + y) / (y + y);
}

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

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

def code(x, y):
	return (x + y) / (y + y)

function code(x, y)
	return Float64(Float64(x + y) / Float64(y + y))
end

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

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(x + y) / (y + y)
END code
\frac{x + y}{y + y}

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\mathsf{fma}\left(\frac{x}{y}, 0.5, 0.5\right) \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fma (/ x y) 0.5 0.5))
double code(double x, double y) {
	return fma((x / y), 0.5, 0.5);
}

function code(x, y)
	return fma(Float64(x / y), 0.5, 0.5)
end

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	((x / y) * (5e-1)) + (5e-1)
END code
\mathsf{fma}\left(\frac{x}{y}, 0.5, 0.5\right)
Derivation

  1. Initial program 99.9%

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

    1. Applied rewrites100.0%

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

    Alternative 2: 97.7% accurate, 0.3× speedup?

    \[\begin{array}{l} t_0 := \frac{x + y}{y + y}\\ t_1 := \frac{x}{y + y}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]
    (FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (/ (+ x y) (+ y y))) (t_1 (/ x (+ y y))))
  (if (<= t_0 -5.0) t_1 (if (<= t_0 1.0) 0.5 t_1))))
    double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double t_1 = x / (y + y);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

    real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + y) / (y + y)
    t_1 = x / (y + y)
    if (t_0 <= (-5.0d0)) then
        tmp = t_1
    else if (t_0 <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

    public static double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double t_1 = x / (y + y);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

    def code(x, y):
	t_0 = (x + y) / (y + y)
	t_1 = x / (y + y)
	tmp = 0
	if t_0 <= -5.0:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = 0.5
	else:
		tmp = t_1
	return tmp

    function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + y))
	t_1 = Float64(x / Float64(y + y))
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	return tmp
end

    function tmp_2 = code(x, y)
	t_0 = (x + y) / (y + y);
	t_1 = x / (y + y);
	tmp = 0.0;
	if (t_0 <= -5.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

    code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], t$95$1, If[LessEqual[t$95$0, 1.0], 0.5, t$95$1]]]]

    f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET t_0 = ((x + y) / (y + y)) IN
		LET t_1 = (x / (y + y)) IN
			LET tmp_1 = IF (t_0 <= (1)) THEN (5e-1) ELSE t_1 ENDIF IN
			LET tmp = IF (t_0 <= (-5)) THEN t_1 ELSE tmp_1 ENDIF IN
	tmp
END code
    \begin{array}{l}
t_0 := \frac{x + y}{y + y}\\
t_1 := \frac{x}{y + y}\\
\mathbf{if}\;t\_0 \leq -5:\\
\;\;\;\;t\_1\\

\mathbf{elif}\;t\_0 \leq 1:\\
\;\;\;\;0.5\\

\mathbf{else}:\\
\;\;\;\;t\_1\\


\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if (/.f64 (+.f64 x y) (+.f64 y y)) < -5 or 1 < (/.f64 (+.f64 x y) (+.f64 y y))

      1. Initial program 99.9%

        \[\frac{x + y}{y + y} \]

      2. Taylor expanded in x around inf

        \[\leadsto \frac{1}{2} \cdot \frac{x}{y} \]
      3. Step-by-step derivation

        1. Applied rewrites49.5%

          \[\leadsto 0.5 \cdot \frac{x}{y} \]
        2. Step-by-step derivation

          1. Applied rewrites49.5%

            \[\leadsto \frac{x}{y + y} \]

          if -5 < (/.f64 (+.f64 x y) (+.f64 y y)) < 1

          1. Initial program 99.9%

            \[\frac{x + y}{y + y} \]

          2. Taylor expanded in x around 0

            \[\leadsto \frac{1}{2} \]
          3. Step-by-step derivation

            1. Applied rewrites51.6%

              \[\leadsto 0.5 \]
          4. Recombined 2 regimes into one program.
          5. Add Preprocessing

          Alternative 3: 51.6% accurate, 10.0× speedup?

          \[0.5 \]
          (FPCore (x y)
  :precision binary64
  :pre TRUE
  0.5)
          double code(double x, double y) {
	return 0.5;
}

          real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0
end function

          public static double code(double x, double y) {
	return 0.5;
}

          def code(x, y):
	return 0.5

          function code(x, y)
	return 0.5
end

          function tmp = code(x, y)
	tmp = 0.5;
end

          code[x_, y_] := 0.5

          f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	5e-1
END code
          0.5
          Derivation

          1. Initial program 99.9%

            \[\frac{x + y}{y + y} \]

          2. Taylor expanded in x around 0

            \[\leadsto \frac{1}{2} \]
          3. Step-by-step derivation

            1. Applied rewrites51.6%

              \[\leadsto 0.5 \]
            2. Add Preprocessing

            Reproduce

            ?
            herbie shell --seed 2026092 
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"
  :precision binary64
  (/ (+ x y) (+ y y)))