Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 11

Speedup: 1.7×

• 71.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
       (t_1 (* (* t_0 (fabs x)) (fabs x))))
  (fabs
   (*
    (/ 1.0 (sqrt PI))
    (+
     (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
     (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
       (t_1 (* (* t_0 (fabs x)) (fabs x))))
  (fabs
   (*
    (/ 1.0 (sqrt PI))
    (+
     (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
     (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (fabs
 (*
  (/ 1.0 (sqrt PI))
  (fma
   (pow (fabs x) 7.0)
   0.047619047619047616
   (fma
    (* 0.2 (fabs x))
    (* (* (* x x) x) x)
    (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ 0.5641895835477563 \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(t\_0 \cdot t\_0\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   0.5641895835477563
   (fabs
    (fma
     (fabs x)
     (fma
      (* 0.2 (* x x))
      (* x x)
      (* (* t_0 t_0) 0.047619047619047616))
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return 0.5641895835477563 * fabs(fma(fabs(x), fma((0.2 * (x * x)), (x * x), ((t_0 * t_0) * 0.047619047619047616)), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(0.5641895835477563 * abs(fma(abs(x), fma(Float64(0.2 * Float64(x * x)), Float64(x * x), Float64(Float64(t_0 * t_0) * 0.047619047619047616)), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(0.5641895835477563 * N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

4. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|x \cdot \left(\mathsf{fma}\left(t\_0, \mathsf{fma}\left(t\_0, 0.047619047619047616, 0.2 \cdot x\right), 0.6666666666666666 \cdot \left(x \cdot x\right)\right) - -2\right)\right| \cdot 0.5641895835477563 \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (fabs
    (*
     x
     (-
      (fma
       t_0
       (fma t_0 0.047619047619047616 (* 0.2 x))
       (* 0.6666666666666666 (* x x)))
      -2.0)))
   0.5641895835477563)))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return fabs((x * (fma(t_0, fma(t_0, 0.047619047619047616, (0.2 * x)), (0.6666666666666666 * (x * x))) - -2.0))) * 0.5641895835477563;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(abs(Float64(x * Float64(fma(t_0, fma(t_0, 0.047619047619047616, Float64(0.2 * x)), Float64(0.6666666666666666 * Float64(x * x))) - -2.0))) * 0.5641895835477563)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Abs[N[(x * N[(N[(t$95$0 * N[(t$95$0 * 0.047619047619047616 + N[(0.2 * x), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

4. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \cdot 0.5641895835477563} \]

8. Applied rewrites 99.8%

   \[\leadsto \left|x \cdot \color{blue}{\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.047619047619047616, 0.2 \cdot x\right), 0.6666666666666666 \cdot \left(x \cdot x\right)\right) - -2\right)}\right| \cdot 0.5641895835477563 \]
9. Add Preprocessing

Alternative 4: 99.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|\mathsf{fma}\left(t\_0, \mathsf{fma}\left(t\_0, 0.047619047619047616, 0.2 \cdot x\right), \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x\right| \cdot 0.5641895835477563 \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (fabs
    (*
     (fma
      t_0
      (fma t_0 0.047619047619047616 (* 0.2 x))
      (fma 0.6666666666666666 (* x x) 2.0))
     x))
   0.5641895835477563)))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return fabs((fma(t_0, fma(t_0, 0.047619047619047616, (0.2 * x)), fma(0.6666666666666666, (x * x), 2.0)) * x)) * 0.5641895835477563;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(abs(Float64(fma(t_0, fma(t_0, 0.047619047619047616, Float64(0.2 * x)), fma(0.6666666666666666, Float64(x * x), 2.0)) * x)) * 0.5641895835477563)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Abs[N[(N[(t$95$0 * N[(t$95$0 * 0.047619047619047616 + N[(0.2 * x), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

4. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \cdot 0.5641895835477563} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\color{blue}{x \cdot \mathsf{fma}\left(\frac{1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5}\right) \cdot x, x, \mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right)\right)\right)}\right| \cdot \frac{5081767996463981}{9007199254740992} \]

   2. *-commutative N/A

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5}\right) \cdot x, x, \mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right)\right)\right) \cdot x}\right| \cdot \frac{5081767996463981}{9007199254740992} \]

   3. lower-*.f64 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right) \cdot x}\right| \cdot 0.5641895835477563 \]

8. Applied rewrites 99.8%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.047619047619047616, 0.2 \cdot x\right), \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}\right| \cdot 0.5641895835477563 \]
9. Add Preprocessing

Alternative 5: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2050000:\\ \;\;\;\;\left|\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{2}\right)\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\left|\left|x\right| \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right| \cdot 0.5641895835477563\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 2050000.0)
  (*
   (fabs
    (* (fabs x) (+ 2.0 (* 0.6666666666666666 (pow (fabs x) 2.0)))))
   0.5641895835477563)
  (*
   (fabs (* (fabs x) (* 0.047619047619047616 (pow (fabs x) 6.0))))
   0.5641895835477563)))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2050000.0) {
		tmp = fabs((fabs(x) * (2.0 + (0.6666666666666666 * pow(fabs(x), 2.0))))) * 0.5641895835477563;
	} else {
		tmp = fabs((fabs(x) * (0.047619047619047616 * pow(fabs(x), 6.0)))) * 0.5641895835477563;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 2050000.0d0) then
        tmp = abs((abs(x) * (2.0d0 + (0.6666666666666666d0 * (abs(x) ** 2.0d0))))) * 0.5641895835477563d0
    else
        tmp = abs((abs(x) * (0.047619047619047616d0 * (abs(x) ** 6.0d0)))) * 0.5641895835477563d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2050000.0) {
		tmp = Math.abs((Math.abs(x) * (2.0 + (0.6666666666666666 * Math.pow(Math.abs(x), 2.0))))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((Math.abs(x) * (0.047619047619047616 * Math.pow(Math.abs(x), 6.0)))) * 0.5641895835477563;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2050000.0:
		tmp = math.fabs((math.fabs(x) * (2.0 + (0.6666666666666666 * math.pow(math.fabs(x), 2.0))))) * 0.5641895835477563
	else:
		tmp = math.fabs((math.fabs(x) * (0.047619047619047616 * math.pow(math.fabs(x), 6.0)))) * 0.5641895835477563
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2050000.0)
		tmp = Float64(abs(Float64(abs(x) * Float64(2.0 + Float64(0.6666666666666666 * (abs(x) ^ 2.0))))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(abs(x) * Float64(0.047619047619047616 * (abs(x) ^ 6.0)))) * 0.5641895835477563);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2050000.0)
		tmp = abs((abs(x) * (2.0 + (0.6666666666666666 * (abs(x) ^ 2.0))))) * 0.5641895835477563;
	else
		tmp = abs((abs(x) * (0.047619047619047616 * (abs(x) ^ 6.0)))) * 0.5641895835477563;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2050000.0], N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.05e6

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

   4. Evaluated real constant 99.8%

      \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \cdot 0.5641895835477563} \]

   7. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{x \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)}\right| \cdot 0.5641895835477563 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|x \cdot \color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right)}\right| \cdot \frac{5081767996463981}{9007199254740992} \]

      2. lower-+.f64 N/A

         \[\leadsto \left|x \cdot \left(2 + \color{blue}{\frac{2}{3} \cdot {x}^{2}}\right)\right| \cdot \frac{5081767996463981}{9007199254740992} \]

      3. lower-*.f64 N/A

         \[\leadsto \left|x \cdot \left(2 + \frac{2}{3} \cdot \color{blue}{{x}^{2}}\right)\right| \cdot \frac{5081767996463981}{9007199254740992} \]

      4. lower-pow.f64 89.3%

         \[\leadsto \left|x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right| \cdot 0.5641895835477563 \]

   9. Applied rewrites 89.3%

      \[\leadsto \left|\color{blue}{x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)}\right| \cdot 0.5641895835477563 \]

   if 2.05e6 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

   4. Evaluated real constant 99.8%

      \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \cdot 0.5641895835477563} \]

   7. Taylor expanded in x around inf

      \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right)}\right| \cdot 0.5641895835477563 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|x \cdot \left(\frac{1}{21} \cdot \color{blue}{{x}^{6}}\right)\right| \cdot \frac{5081767996463981}{9007199254740992} \]

      2. lower-pow.f64 36.6%

         \[\leadsto \left|x \cdot \left(0.047619047619047616 \cdot {x}^{\color{blue}{6}}\right)\right| \cdot 0.5641895835477563 \]

   9. Applied rewrites 36.6%

      \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right)}\right| \cdot 0.5641895835477563 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 6000000:\\ \;\;\;\;\left|\left(-\left|x\right|\right) \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left|x\right| \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right| \cdot 0.5641895835477563\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 6000000.0)
  (fabs (* (- (fabs x)) 1.1283791670955126))
  (*
   (fabs (* (fabs x) (* 0.047619047619047616 (pow (fabs x) 6.0))))
   0.5641895835477563)))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 6000000.0) {
		tmp = fabs((-fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((fabs(x) * (0.047619047619047616 * pow(fabs(x), 6.0)))) * 0.5641895835477563;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 6000000.0d0) then
        tmp = abs((-abs(x) * 1.1283791670955126d0))
    else
        tmp = abs((abs(x) * (0.047619047619047616d0 * (abs(x) ** 6.0d0)))) * 0.5641895835477563d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 6000000.0) {
		tmp = Math.abs((-Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((Math.abs(x) * (0.047619047619047616 * Math.pow(Math.abs(x), 6.0)))) * 0.5641895835477563;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 6000000.0:
		tmp = math.fabs((-math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((math.fabs(x) * (0.047619047619047616 * math.pow(math.fabs(x), 6.0)))) * 0.5641895835477563
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 6000000.0)
		tmp = abs(Float64(Float64(-abs(x)) * 1.1283791670955126));
	else
		tmp = Float64(abs(Float64(abs(x) * Float64(0.047619047619047616 * (abs(x) ^ 6.0)))) * 0.5641895835477563);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 6000000.0)
		tmp = abs((-abs(x) * 1.1283791670955126));
	else
		tmp = abs((abs(x) * (0.047619047619047616 * (abs(x) ^ 6.0)))) * 0.5641895835477563;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 6000000.0], N[Abs[N[((-N[Abs[x], $MachinePrecision]) * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e6

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.4%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)}}}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      2. lower-/.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. lower-fabs.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-PI.f64 67.4%

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 67.4%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]

   7. Evaluated real constant 67.7%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. lift-fabs.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. rem-sqrt-square-rev N/A

          \[\leadsto \left|\sqrt{x \cdot x} \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      11. sqr-neg-rev N/A

          \[\leadsto \left|\sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      12. lift-neg.f64 N/A

          \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      13. lift-neg.f64 N/A

          \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      14. sqrt-unprod N/A

          \[\leadsto \left|\left(\sqrt{-x} \cdot \sqrt{-x}\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. rem-square-sqrt N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      16. metadata-eval N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      17. metadata-eval N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      18. metadata-eval 67.8%

          \[\leadsto \left|\left(-x\right) \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.8%

      \[\leadsto \left|\left(-x\right) \cdot \color{blue}{1.1283791670955126}\right| \]

   if 6e6 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

   4. Evaluated real constant 99.8%

      \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \cdot 0.5641895835477563} \]

   7. Taylor expanded in x around inf

      \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right)}\right| \cdot 0.5641895835477563 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|x \cdot \left(\frac{1}{21} \cdot \color{blue}{{x}^{6}}\right)\right| \cdot \frac{5081767996463981}{9007199254740992} \]

      2. lower-pow.f64 36.6%

         \[\leadsto \left|x \cdot \left(0.047619047619047616 \cdot {x}^{\color{blue}{6}}\right)\right| \cdot 0.5641895835477563 \]

   9. Applied rewrites 36.6%

      \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right)}\right| \cdot 0.5641895835477563 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.5% accurate, 2.5× speedup

Math

\[\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (fabs
 (*
  (/ 1.0 (sqrt PI))
  (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x)))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x)))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)}\right| \]
5. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{1}{21}, \color{blue}{{\left(\left|x\right|\right)}^{7}}, 2 \cdot \left|x\right|\right)\right| \]

   2. lower-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{\color{blue}{7}}, 2 \cdot \left|x\right|\right)\right| \]

   3. lower-fabs.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right| \]

   5. lower-fabs.f64 98.7%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right| \]

6. Applied rewrites 98.7%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}\right| \]
7. Add Preprocessing

Alternative 8: 98.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 6000000:\\ \;\;\;\;\left|\left(-\left|x\right|\right) \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right| \cdot 0.5641895835477563\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 6000000.0)
  (fabs (* (- (fabs x)) 1.1283791670955126))
  (*
   (fabs (* 0.047619047619047616 (pow (fabs x) 7.0)))
   0.5641895835477563)))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 6000000.0) {
		tmp = fabs((-fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((0.047619047619047616 * pow(fabs(x), 7.0))) * 0.5641895835477563;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 6000000.0d0) then
        tmp = abs((-abs(x) * 1.1283791670955126d0))
    else
        tmp = abs((0.047619047619047616d0 * (abs(x) ** 7.0d0))) * 0.5641895835477563d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 6000000.0) {
		tmp = Math.abs((-Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.pow(Math.abs(x), 7.0))) * 0.5641895835477563;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 6000000.0:
		tmp = math.fabs((-math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((0.047619047619047616 * math.pow(math.fabs(x), 7.0))) * 0.5641895835477563
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 6000000.0)
		tmp = abs(Float64(Float64(-abs(x)) * 1.1283791670955126));
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * (abs(x) ^ 7.0))) * 0.5641895835477563);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 6000000.0)
		tmp = abs((-abs(x) * 1.1283791670955126));
	else
		tmp = abs((0.047619047619047616 * (abs(x) ^ 7.0))) * 0.5641895835477563;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 6000000.0], N[Abs[N[((-N[Abs[x], $MachinePrecision]) * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e6

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.4%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)}}}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      2. lower-/.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. lower-fabs.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-PI.f64 67.4%

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 67.4%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]

   7. Evaluated real constant 67.7%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. lift-fabs.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. rem-sqrt-square-rev N/A

          \[\leadsto \left|\sqrt{x \cdot x} \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      11. sqr-neg-rev N/A

          \[\leadsto \left|\sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      12. lift-neg.f64 N/A

          \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      13. lift-neg.f64 N/A

          \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      14. sqrt-unprod N/A

          \[\leadsto \left|\left(\sqrt{-x} \cdot \sqrt{-x}\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. rem-square-sqrt N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      16. metadata-eval N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      17. metadata-eval N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      18. metadata-eval 67.8%

          \[\leadsto \left|\left(-x\right) \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.8%

      \[\leadsto \left|\left(-x\right) \cdot \color{blue}{1.1283791670955126}\right| \]

   if 6e6 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right|} \]

   4. Evaluated real constant 99.8%

      \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right), x, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot x, x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \cdot 0.5641895835477563} \]

   7. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot {x}^{7}}\right| \cdot 0.5641895835477563 \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{{x}^{7}}\right| \cdot \frac{5081767996463981}{9007199254740992} \]

      2. lower-pow.f64 36.6%

         \[\leadsto \left|0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right| \cdot 0.5641895835477563 \]

   9. Applied rewrites 36.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot {x}^{7}}\right| \cdot 0.5641895835477563 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.4% accurate, 2.9× speedup

Math

\[\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (fabs
 (/
  (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x)))
  1.772453850905516)))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x))) / 1.772453850905516));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x))) / 1.772453850905516))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left|\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   2. lower-fma.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

   3. lower-pow.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   4. lower-fabs.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   6. lower-fabs.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   8. lower-PI.f64 98.3%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

6. Applied rewrites 98.3%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}}\right| \]

7. Evaluated real constant 98.5%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]
8. Add Preprocessing

Alternative 10: 83.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|\left(-\left|x\right|\right) \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{\left|x\right| \cdot \left|x\right|}{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 0.05)
  (fabs (* (- (fabs x)) 1.1283791670955126))
  (fabs (* 2.0 (sqrt (/ (* (fabs x) (fabs x)) PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((-fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * sqrt(((fabs(x) * fabs(x)) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.05) {
		tmp = Math.abs((-Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((Math.abs(x) * Math.abs(x)) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.05:
		tmp = math.fabs((-math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((math.fabs(x) * math.fabs(x)) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(Float64(-abs(x)) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(abs(x) * abs(x)) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.05)
		tmp = abs((-abs(x) * 1.1283791670955126));
	else
		tmp = abs((2.0 * sqrt(((abs(x) * abs(x)) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[((-N[Abs[x], $MachinePrecision]) * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.050000000000000003

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.4%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)}}}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      2. lower-/.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. lower-fabs.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-PI.f64 67.4%

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 67.4%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]

   7. Evaluated real constant 67.7%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. lift-fabs.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. rem-sqrt-square-rev N/A

          \[\leadsto \left|\sqrt{x \cdot x} \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      11. sqr-neg-rev N/A

          \[\leadsto \left|\sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      12. lift-neg.f64 N/A

          \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      13. lift-neg.f64 N/A

          \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      14. sqrt-unprod N/A

          \[\leadsto \left|\left(\sqrt{-x} \cdot \sqrt{-x}\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. rem-square-sqrt N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      16. metadata-eval N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      17. metadata-eval N/A

          \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      18. metadata-eval 67.8%

          \[\leadsto \left|\left(-x\right) \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.8%

      \[\leadsto \left|\left(-x\right) \cdot \color{blue}{1.1283791670955126}\right| \]

   if 0.050000000000000003 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.4%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)}}}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      2. lower-/.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. lower-fabs.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-PI.f64 67.4%

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 67.4%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\pi}}}\right| \]

      2. lift-fabs.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\color{blue}{\pi}}}\right| \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \left|2 \cdot \frac{\sqrt{x \cdot x}}{\sqrt{\color{blue}{\pi}}}\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\sqrt{x \cdot x}}{\sqrt{\pi}}\right| \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\sqrt{x \cdot x}}{\sqrt{\pi}}\right| \]

      6. sqrt-undiv N/A

         \[\leadsto \left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right| \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right| \]

      8. lower-/.f64 53.3%

         \[\leadsto \left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right| \]

   8. Applied rewrites 53.3%

      \[\leadsto \left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.8% accurate, 15.7× speedup

Math

\[\left|\left(-x\right) \cdot 1.1283791670955126\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (fabs (* (- x) 1.1283791670955126)))

C

double code(double x) {
	return fabs((-x * 1.1283791670955126));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = abs((-x * 1.1283791670955126d0))
end function

Java

public static double code(double x) {
	return Math.abs((-x * 1.1283791670955126));
}

Python

def code(x):
	return math.fabs((-x * 1.1283791670955126))

Julia

function code(x)
	return abs(Float64(Float64(-x) * 1.1283791670955126))
end

MATLAB

function tmp = code(x)
	tmp = abs((-x * 1.1283791670955126));
end

Wolfram

code[x_] := N[Abs[N[((-x) * 1.1283791670955126), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.4%

   \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)}}}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   2. lower-/.f64 N/A

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   3. lower-fabs.f64 N/A

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   5. lower-PI.f64 67.4%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right| \]

6. Applied rewrites 67.4%

   \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]

7. Evaluated real constant 67.7%

   \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   2. count-2-rev N/A

      \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   3. lift-/.f64 N/A

      \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   4. mult-flip N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   5. lift-/.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   6. mult-flip N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   7. distribute-lft-out N/A

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   8. lower-*.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   9. lift-fabs.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   10. rem-sqrt-square-rev N/A

       \[\leadsto \left|\sqrt{x \cdot x} \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   11. sqr-neg-rev N/A

       \[\leadsto \left|\sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   12. lift-neg.f64 N/A

       \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   13. lift-neg.f64 N/A

       \[\leadsto \left|\sqrt{\left(-x\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   14. sqrt-unprod N/A

       \[\leadsto \left|\left(\sqrt{-x} \cdot \sqrt{-x}\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   15. rem-square-sqrt N/A

       \[\leadsto \left|\left(-x\right) \cdot \left(\color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   16. metadata-eval N/A

       \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   17. metadata-eval N/A

       \[\leadsto \left|\left(-x\right) \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

   18. metadata-eval 67.8%

       \[\leadsto \left|\left(-x\right) \cdot 1.1283791670955126\right| \]

9. Applied rewrites 67.8%

   \[\leadsto \left|\left(-x\right) \cdot \color{blue}{1.1283791670955126}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))