Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 3.6s

Alternatives: 10

Speedup: 1.1×

Specification

Math

\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))
END code

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))
END code

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[-6 \cdot \frac{1 - x}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (* -6.0 (/ (- 1.0 x) (fma (sqrt x) 4.0 (+ 1.0 x)))))

C

double code(double x) {
	return -6.0 * ((1.0 - x) / fma(sqrt(x), 4.0, (1.0 + x)));
}

Julia

function code(x)
	return Float64(-6.0 * Float64(Float64(1.0 - x) / fma(sqrt(x), 4.0, Float64(1.0 + x))))
end

Wolfram

code[x_] := N[(-6.0 * N[(N[(1.0 - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(-6) * (((1) - x) / (((sqrt(x)) * (4)) + ((1) + x)))
END code

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

3. Applied rewrites 99.9%

   \[\leadsto -6 \cdot \frac{1 - x}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)} \]
4. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\frac{\mathsf{fma}\left(-6, x, 6\right)}{\mathsf{fma}\left(-4, \sqrt{x}, -1 - x\right)} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (/ (fma -6.0 x 6.0) (fma -4.0 (sqrt x) (- -1.0 x))))

C

double code(double x) {
	return fma(-6.0, x, 6.0) / fma(-4.0, sqrt(x), (-1.0 - x));
}

Julia

function code(x)
	return Float64(fma(-6.0, x, 6.0) / fma(-4.0, sqrt(x), Float64(-1.0 - x)))
end

Wolfram

code[x_] := N[(N[(-6.0 * x + 6.0), $MachinePrecision] / N[(-4.0 * N[Sqrt[x], $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(((-6) * x) + (6)) / (((-4) * (sqrt(x))) + ((-1) - x))
END code

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Step-by-step derivation

3. Applied rewrites 99.7%

   \[\leadsto \frac{\mathsf{fma}\left(-6, x, 6\right)}{\mathsf{fma}\left(-4, \sqrt{x}, -1 - x\right)} \]
4. Add Preprocessing

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := 6 \cdot \left(x - 1\right)\\ \mathbf{if}\;\frac{t\_0}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{4}{\sqrt{x}} - -1}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* 6.0 (- x 1.0))))
  (if (<= (/ t_0 (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
    (/ t_0 (fma 4.0 (sqrt x) 1.0))
    (/ 6.0 (- (/ 4.0 (sqrt x)) -1.0)))))

C

double code(double x) {
	double t_0 = 6.0 * (x - 1.0);
	double tmp;
	if ((t_0 / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = t_0 / fma(4.0, sqrt(x), 1.0);
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(6.0 * Float64(x - 1.0))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(t_0 / fma(4.0, sqrt(x), 1.0));
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(t$95$0 / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET t_0 = ((6) * (x - (1))) IN
		LET tmp = IF ((t_0 / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN (t_0 / (((4) * (sqrt(x))) + (1))) ELSE ((6) / (((4) / (sqrt(x))) - (-1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 + 4 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.7%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 + 4 \cdot \sqrt{x}} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.7%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   6. Applied rewrites 50.4%

      \[\leadsto \frac{6}{\frac{4}{\sqrt{x}} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{4}{\sqrt{x}} - -1}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
  (/ (fma x 6.0 -6.0) (fma (sqrt x) 4.0 1.0))
  (/ 6.0 (- (/ 4.0 (sqrt x)) -1.0))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(fma(x, 6.0, -6.0) / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN (((x * (6)) + (-6)) / (((sqrt(x)) * (4)) + (1))) ELSE ((6) / (((4) / (sqrt(x))) - (-1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 + 4 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.7%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{1 + 4 \cdot \sqrt{x}} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.7%

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   6. Applied rewrites 50.4%

      \[\leadsto \frac{6}{\frac{4}{\sqrt{x}} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(-4, \sqrt{x}, -1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{4}{\sqrt{x}} - -1}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
  (/ 6.0 (fma -4.0 (sqrt x) (- -1.0 x)))
  (/ 6.0 (- (/ 4.0 (sqrt x)) -1.0))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = 6.0 / fma(-4.0, sqrt(x), (-1.0 - x));
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(6.0 / fma(-4.0, sqrt(x), Float64(-1.0 - x)));
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(6.0 / N[(-4.0 * N[Sqrt[x], $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN ((6) / (((-4) * (sqrt(x))) + ((-1) - x))) ELSE ((6) / (((4) / (sqrt(x))) - (-1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(-6, x, 6\right)}{\mathsf{fma}\left(-4, \sqrt{x}, -1 - x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{6}{\mathsf{fma}\left(-4, \sqrt{x}, -1 - x\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 49.3%

      \[\leadsto \frac{6}{\mathsf{fma}\left(-4, \sqrt{x}, -1 - x\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   6. Applied rewrites 50.4%

      \[\leadsto \frac{6}{\frac{4}{\sqrt{x}} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt{x}, -0.6666666666666666, -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{\frac{4}{\sqrt{x}} - -1}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
  (/ 1.0 (fma (sqrt x) -0.6666666666666666 -0.16666666666666666))
  (/ 6.0 (- (/ 4.0 (sqrt x)) -1.0))))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = 1.0 / fma(sqrt(x), -0.6666666666666666, -0.16666666666666666);
	} else {
		tmp = 6.0 / ((4.0 / sqrt(x)) - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(1.0 / fma(sqrt(x), -0.6666666666666666, -0.16666666666666666));
	else
		tmp = Float64(6.0 / Float64(Float64(4.0 / sqrt(x)) - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * -0.6666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(N[(4.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN ((1) / (((sqrt(x)) * (-66666666666666662965923251249478198587894439697265625e-53)) + (-1666666666666666574148081281236954964697360992431640625e-55))) ELSE ((6) / (((4) / (sqrt(x))) - (-1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.0%

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{x}, -0.6666666666666666, -0.16666666666666666\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   6. Applied rewrites 50.4%

      \[\leadsto \frac{6}{\frac{4}{\sqrt{x}} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt{x}, -0.6666666666666666, -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
  (/ 1.0 (fma (sqrt x) -0.6666666666666666 -0.16666666666666666))
  6.0))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = 1.0 / fma(sqrt(x), -0.6666666666666666, -0.16666666666666666);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(1.0 / fma(sqrt(x), -0.6666666666666666, -0.16666666666666666));
	else
		tmp = 6.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] * -0.6666666666666666 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision], 6.0]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN ((1) / (((sqrt(x)) * (-66666666666666662965923251249478198587894439697265625e-53)) + (-1666666666666666574148081281236954964697360992431640625e-55))) ELSE (6) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.0%

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{x}, -0.6666666666666666, -0.16666666666666666\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \sqrt{x} \]

   10. Taylor expanded in x around inf

       \[\leadsto 6 \]
   11. Step-by-step derivation

   12. Applied rewrites 49.1%

       \[\leadsto 6 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
  (/ -6.0 (fma (sqrt x) 4.0 1.0))
  6.0))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, 1.0);
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, 1.0));
	else
		tmp = 6.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision], 6.0]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN ((-6) / (((sqrt(x)) * (4)) + (1))) ELSE (6) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.0%

      \[\leadsto \frac{-6}{1 + 4 \cdot \sqrt{x}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right)} \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \sqrt{x} \]

   10. Taylor expanded in x around inf

       \[\leadsto 6 \]
   11. Step-by-step derivation

   12. Applied rewrites 49.1%

       \[\leadsto 6 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -1:\\ \;\;\;\;\sqrt{x} \cdot -1.5\\ \mathbf{else}:\\ \;\;\;\;6\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -1.0)
  (* (sqrt x) -1.5)
  6.0))

C

double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0) {
		tmp = sqrt(x) * -1.5;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))) <= (-1.0d0)) then
        tmp = sqrt(x) * (-1.5d0)
    else
        tmp = 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)))) <= -1.0) {
		tmp = Math.sqrt(x) * -1.5;
	} else {
		tmp = 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))) <= -1.0:
		tmp = math.sqrt(x) * -1.5
	else:
		tmp = 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -1.0)
		tmp = Float64(sqrt(x) * -1.5);
	else
		tmp = 6.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -1.0)
		tmp = sqrt(x) * -1.5;
	else
		tmp = 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[Sqrt[x], $MachinePrecision] * -1.5), $MachinePrecision], 6.0]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((((6) * (x - (1))) / ((x + (1)) + ((4) * (sqrt(x))))) <= (-1)) THEN ((sqrt(x)) * (-15e-1)) ELSE (6) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -1

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]

   8. Taylor expanded in x around -inf

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

   10. Applied rewrites 4.1%

       \[\leadsto \frac{-1.5}{\sqrt{\frac{1}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 4.1%

       \[\leadsto \sqrt{x} \cdot -1.5 \]

   if -1 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.4%

      \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.5%

      \[\leadsto 1.5 \cdot \sqrt{x} \]

   10. Taylor expanded in x around inf

       \[\leadsto 6 \]
   11. Step-by-step derivation

   12. Applied rewrites 49.1%

       \[\leadsto 6 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.1% accurate, 20.6× speedup

Math

\[6 \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	6
END code

Derivation

1. Initial program 99.7%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

2. Taylor expanded in x around inf

   \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]
3. Step-by-step derivation

4. Applied rewrites 50.4%

   \[\leadsto \frac{6}{1 + 4 \cdot \sqrt{\frac{1}{x}}} \]

5. Taylor expanded in x around 0

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

7. Applied rewrites 4.5%

   \[\leadsto 1.5 \cdot \frac{x}{\sqrt{x}} \]
8. Step-by-step derivation

9. Applied rewrites 4.5%

   \[\leadsto 1.5 \cdot \sqrt{x} \]

10. Taylor expanded in x around inf

    \[\leadsto 6 \]
11. Step-by-step derivation

12. Applied rewrites 49.1%

    \[\leadsto 6 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64
  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))