Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 1.1s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\frac{x}{1 - x} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (/ x (- 1.0 x)))

C

double code(double x) {
	return x / (1.0 - x);
}

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 - x);
}

Python

def code(x):
	return x / (1.0 - x)

Julia

function code(x)
	return Float64(x / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 - x);
end

Wolfram

code[x_] := N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	x / ((1) - x)
END code

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\frac{x}{1 - x} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (/ x (- 1.0 x)))

C

double code(double x) {
	return x / (1.0 - x);
}

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 - x);
}

Python

def code(x):
	return x / (1.0 - x)

Julia

function code(x)
	return Float64(x / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 - x);
end

Wolfram

code[x_] := N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	x / ((1) - x)
END code

Alternative 1: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, x\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (- 1.0 x)) -0.5) -1.0 (fma x x x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = fma(x, x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = fma(x, x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(x * x + x), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) - x)) <= (-5e-1)) THEN (-1) ELSE ((x * x) + x) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -0.5

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 52.1%

      \[\leadsto -1 \]

   if -0.5 < (/.f64 x (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.9%

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

   6. Applied rewrites 48.9%

      \[\leadsto \mathsf{fma}\left(x, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 - x} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (- 1.0 x)) -0.5) -1.0 (/ x 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 - x)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = x / 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 - x)) <= -0.5:
		tmp = -1.0
	else:
		tmp = x / 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(x / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 - x)) <= -0.5)
		tmp = -1.0;
	else
		tmp = x / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -0.5], -1.0, N[(x / 1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) - x)) <= (-5e-1)) THEN (-1) ELSE (x / (1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 #s(literal 1 binary64) x)) < -0.5

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 52.1%

      \[\leadsto -1 \]

   if -0.5 < (/.f64 x (-.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{x}{1 - x} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 49.7%

      \[\leadsto \frac{x}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 52.1% accurate, 7.4× speedup

Math

\[-1 \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	-1
END code

Derivation

1. Initial program 100.0%

   \[\frac{x}{1 - x} \]

2. Taylor expanded in x around inf

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

4. Applied rewrites 52.1%

   \[\leadsto -1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x)
  :name "Numeric.Integration.TanhSinh:nonNegative from integration-0.2.1"
  :precision binary64
  (/ x (- 1.0 x)))