Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.5% → 99.9%
Time: 1.5s
Alternatives: 3
Speedup: 1.1×

Specification

?
\[x \cdot \left(1 + y \cdot y\right) \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x (+ 1.0 (* y y))))
double code(double x, double y) {
	return x * (1.0 + (y * y));
}

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

def code(x, y):
	return x * (1.0 + (y * y))

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * ((1) + (y * y))
END code
x \cdot \left(1 + y \cdot y\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 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: 94.5% accurate, 1.0× speedup?

\[x \cdot \left(1 + y \cdot y\right) \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x (+ 1.0 (* y y))))
double code(double x, double y) {
	return x * (1.0 + (y * y));
}

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

def code(x, y):
	return x * (1.0 + (y * y))

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * ((1) + (y * y))
END code
x \cdot \left(1 + y \cdot y\right)

Alternative 1: 99.9% accurate, 1.1× speedup?

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

function code(x, y)
	return fma(y, Float64(y * x), x)
end

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

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

  1. Initial program 94.5%

    \[x \cdot \left(1 + y \cdot y\right) \]
  2. Step-by-step derivation

    1. Applied rewrites99.9%

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

    Alternative 2: 94.5% accurate, 1.1× speedup?

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

    function code(x, y)
	return fma(x, Float64(y * y), x)
end

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

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

    1. Initial program 94.5%

      \[x \cdot \left(1 + y \cdot y\right) \]
    2. Step-by-step derivation

      1. Applied rewrites94.5%

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

      Alternative 3: 51.9% accurate, 2.4× speedup?

      \[x \cdot 1 \]
      (FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x 1.0))
      double code(double x, double y) {
	return x * 1.0;
}

      real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 1.0d0
end function

      public static double code(double x, double y) {
	return x * 1.0;
}

      def code(x, y):
	return x * 1.0

      function code(x, y)
	return Float64(x * 1.0)
end

      function tmp = code(x, y)
	tmp = x * 1.0;
end

      code[x_, y_] := N[(x * 1.0), $MachinePrecision]

      f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * (1)
END code
      x \cdot 1
      Derivation

      1. Initial program 94.5%

        \[x \cdot \left(1 + y \cdot y\right) \]

      2. Taylor expanded in y around 0

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

        1. Applied rewrites51.9%

          \[\leadsto x \cdot 1 \]
        2. Add Preprocessing

        Reproduce

        ?
        herbie shell --seed 2026092 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64
  (* x (+ 1.0 (* y y))))