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

Percentage Accurate: 99.8% → 99.8%

Time: 4.7s

Alternatives: 11

Speedup: 2.2×

• 28.6% 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

\[\begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(t\_0 \cdot \left|x\right|, \left(x \cdot x\right) \cdot 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, t\_0, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (fma
      (* t_0 (fabs x))
      (* (* x x) 0.047619047619047616)
      (fma
       (* 0.2 (fabs x))
       t_0
       (* (fabs x) (fma (* x x) 0.6666666666666666 2.0))))))))

C

double code(double x) {
	double t_0 = ((x * x) * x) * x;
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma((t_0 * fabs(x)), ((x * x) * 0.047619047619047616), fma((0.2 * fabs(x)), t_0, (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(Float64(t_0 * abs(x)), Float64(Float64(x * x) * 0.047619047619047616), fma(Float64(0.2 * abs(x)), t_0, Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 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%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(t\_0 \cdot t\_0\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (/ 1.0 (sqrt PI))
    (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 (1.0 / sqrt(((double) M_PI))) * 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(Float64(1.0 / sqrt(pi)) * 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[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 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. Add Preprocessing

Alternative 3: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(N[(0.047619047619047616 * x), $MachinePrecision] * x + 0.2), $MachinePrecision] * N[(N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $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%

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

5. Applied rewrites 99.8%

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

6. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{\frac{5081767996463981}{9007199254740992}} \cdot \left|\mathsf{fma}\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \mathsf{fma}\left(\frac{1}{21} \cdot x, x, \frac{1}{5}\right), \mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right) \cdot \left|x\right|\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 4: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(0.5641895835477563 * N[Abs[N[(N[(N[(0.047619047619047616 * x), $MachinePrecision] * x + 0.2), $MachinePrecision] * N[(N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * 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%

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

5. Applied rewrites 99.8%

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

6. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{\frac{5081767996463981}{9007199254740992}} \cdot \left|\mathsf{fma}\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \mathsf{fma}\left(\frac{1}{21} \cdot x, x, \frac{1}{5}\right), \mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right) \cdot \left|x\right|\right)\right| \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. lift-fabs.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

8. Applied rewrites 99.8%

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

Alternative 5: 99.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(0.5641895835477563 * N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * x), $MachinePrecision] * x + 0.2), $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%

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

5. Applied rewrites 99.8%

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

6. Evaluated real constant 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \left|\mathsf{fma}\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \mathsf{fma}\left(\frac{1}{21} \cdot x, x, \frac{1}{5}\right), \color{blue}{2} \cdot \left|x\right|\right)\right| \]
8. Step-by-step derivation

9. Applied rewrites 99.0%

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

Alternative 6: 98.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 0.32:\\ \;\;\;\;\left|t\_0 \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{t\_0}^{7} \cdot 0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 0.32)
     (fabs (* t_0 1.1283791670955126))
     (fabs (/ (* (pow t_0 7.0) 0.047619047619047616) (sqrt PI))))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 0.32) {
		tmp = fabs((t_0 * 1.1283791670955126));
	} else {
		tmp = fabs(((pow(t_0, 7.0) * 0.047619047619047616) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 0.32) {
		tmp = Math.abs((t_0 * 1.1283791670955126));
	} else {
		tmp = Math.abs(((Math.pow(t_0, 7.0) * 0.047619047619047616) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 0.32:
		tmp = math.fabs((t_0 * 1.1283791670955126))
	else:
		tmp = math.fabs(((math.pow(t_0, 7.0) * 0.047619047619047616) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 0.32)
		tmp = abs(Float64(t_0 * 1.1283791670955126));
	else
		tmp = abs(Float64(Float64((t_0 ^ 7.0) * 0.047619047619047616) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(abs(x));
	tmp = 0.0;
	if (abs(x) <= 0.32)
		tmp = abs((t_0 * 1.1283791670955126));
	else
		tmp = abs((((t_0 ^ 7.0) * 0.047619047619047616) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.32], N[Abs[N[(t$95$0 * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Power[t$95$0, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.320000000000000007

   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%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   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 67.2

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

   6. Applied rewrites 67.2%

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

   7. Evaluated real constant 67.5%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}\right| \]
   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. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 67.7

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.7%

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

   if 0.320000000000000007 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-PI.f64 36.9

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 36.9%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

   8. Applied rewrites 36.9%

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

Alternative 7: 98.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 0.32:\\ \;\;\;\;\left|t\_0 \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{t\_0}^{7}}{\sqrt{\pi}} \cdot 0.047619047619047616\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 0.32)
     (fabs (* t_0 1.1283791670955126))
     (fabs (* (/ (pow t_0 7.0) (sqrt PI)) 0.047619047619047616)))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 0.32) {
		tmp = fabs((t_0 * 1.1283791670955126));
	} else {
		tmp = fabs(((pow(t_0, 7.0) / sqrt(((double) M_PI))) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 0.32) {
		tmp = Math.abs((t_0 * 1.1283791670955126));
	} else {
		tmp = Math.abs(((Math.pow(t_0, 7.0) / Math.sqrt(Math.PI)) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 0.32:
		tmp = math.fabs((t_0 * 1.1283791670955126))
	else:
		tmp = math.fabs(((math.pow(t_0, 7.0) / math.sqrt(math.pi)) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 0.32)
		tmp = abs(Float64(t_0 * 1.1283791670955126));
	else
		tmp = abs(Float64(Float64((t_0 ^ 7.0) / sqrt(pi)) * 0.047619047619047616));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(abs(x));
	tmp = 0.0;
	if (abs(x) <= 0.32)
		tmp = abs((t_0 * 1.1283791670955126));
	else
		tmp = abs((((t_0 ^ 7.0) / sqrt(pi)) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.32], N[Abs[N[(t$95$0 * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Power[t$95$0, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.320000000000000007

   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%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   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 67.2

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

   6. Applied rewrites 67.2%

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

   7. Evaluated real constant 67.5%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}\right| \]
   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. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 67.7

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.7%

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

   if 0.320000000000000007 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      5. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-PI.f64 36.9

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 36.9%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{21}}\right| \]

      3. lower-*.f64 36.9

         \[\leadsto \left|\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \color{blue}{0.047619047619047616}\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \frac{1}{21}\right| \]

      5. lift-pow.f64 N/A

         \[\leadsto \left|\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \frac{1}{21}\right| \]

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

         \[\leadsto \left|\frac{\left({x}^{3} \cdot {x}^{3}\right) \cdot \left|x\right|}{\sqrt{\pi}} \cdot \frac{1}{21}\right| \]

      8. pow-prod-down N/A

         \[\leadsto \left|\frac{{\left(x \cdot x\right)}^{3} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \frac{1}{21}\right| \]

      9. lift-fabs.f64 N/A

         \[\leadsto \left|\frac{{\left(x \cdot x\right)}^{3} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \frac{1}{21}\right| \]

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

          \[\leadsto \left|\frac{{\left(x \cdot x\right)}^{3} \cdot \sqrt{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1}{21}\right| \]

      11. pow1/2 N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow2 N/A

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

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

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

      17. lift-fabs.f64 N/A

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

      18. lift-pow.f64 36.9

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

   8. Applied rewrites 36.9%

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

Alternative 8: 98.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * -0.047619047619047616 + 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 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 98.4

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

6. Applied rewrites 98.4%

   \[\leadsto \left|\frac{\color{blue}{-0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} - 2 \cdot \left|x\right|}}{-\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-fabs.f64 N/A

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

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

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

   8. sqrt-pow2 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

       \[\leadsto \left|\frac{{\left(x \cdot x\right)}^{\left(3 + \frac{1}{2}\right)} \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   11. pow-prod-up N/A

       \[\leadsto \left|\frac{\left({\left(x \cdot x\right)}^{3} \cdot {\left(x \cdot x\right)}^{\frac{1}{2}}\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(\color{blue}{2} \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   12. pow-prod-down N/A

       \[\leadsto \left|\frac{\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{2}}\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   13. pow-prod-up N/A

       \[\leadsto \left|\frac{\left({x}^{\left(3 + 3\right)} \cdot {\left(x \cdot x\right)}^{\frac{1}{2}}\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   14. metadata-eval N/A

       \[\leadsto \left|\frac{\left({x}^{6} \cdot {\left(x \cdot x\right)}^{\frac{1}{2}}\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   15. lift-pow.f64 N/A

       \[\leadsto \left|\frac{\left({x}^{6} \cdot {\left(x \cdot x\right)}^{\frac{1}{2}}\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   16. pow1/2 N/A

       \[\leadsto \left|\frac{\left({x}^{6} \cdot \sqrt{x \cdot x}\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

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

       \[\leadsto \left|\frac{\left({x}^{6} \cdot \left|x\right|\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   18. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{\left({x}^{6} \cdot \left|x\right|\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   19. lift-*.f64 N/A

       \[\leadsto \left|\frac{\left({x}^{6} \cdot \left|x\right|\right) \cdot \frac{-1}{21} + \left(\mathsf{neg}\left(\color{blue}{2} \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

   20. lower-fma.f64 N/A

       \[\leadsto \left|\frac{\mathsf{fma}\left({x}^{6} \cdot \left|x\right|, \color{blue}{\frac{-1}{21}}, \mathsf{neg}\left(2 \cdot \left|x\right|\right)\right)}{-\sqrt{\pi}}\right| \]

8. Applied rewrites 98.4%

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

9. Evaluated real constant 98.6%

   \[\leadsto \left|\frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{-1}{21}, -2 \cdot \left|x\right|\right)}{\color{blue}{\frac{-7982422502469483}{4503599627370496}}}\right| \]
10. Add Preprocessing

Alternative 9: 83.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\left|\left|\left|x\right|\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{\left|x\right| \cdot \left|x\right|}{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e-25)
   (fabs (* (fabs (fabs x)) 1.1283791670955126))
   (fabs (* 2.0 (sqrt (/ (* (fabs x) (fabs x)) PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-25) {
		tmp = fabs((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) <= 5e-25) {
		tmp = Math.abs((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) <= 5e-25:
		tmp = math.fabs((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) <= 5e-25)
		tmp = abs(Float64(abs(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) <= 5e-25)
		tmp = abs((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], 5e-25], N[Abs[N[(N[Abs[N[Abs[x], $MachinePrecision]], $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 < 4.99999999999999962e-25

   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%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   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 67.2

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

   6. Applied rewrites 67.2%

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

   7. Evaluated real constant 67.5%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}\right| \]
   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. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 67.7

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.7%

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

   if 4.99999999999999962e-25 < 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%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   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 67.2

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

   6. Applied rewrites 67.2%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   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-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 10: 67.7% accurate, 10.2× speedup

Math

\[0.5641895835477563 \cdot \left|2 \cdot \left|x\right|\right| \]

FPCore

(FPCore (x) :precision binary64 (* 0.5641895835477563 (fabs (* 2.0 (fabs x)))))

C

double code(double x) {
	return 0.5641895835477563 * fabs((2.0 * fabs(x)));
}

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 = 0.5641895835477563d0 * abs((2.0d0 * abs(x)))
end function

Java

public static double code(double x) {
	return 0.5641895835477563 * Math.abs((2.0 * Math.abs(x)));
}

Python

def code(x):
	return 0.5641895835477563 * math.fabs((2.0 * math.fabs(x)))

Julia

function code(x)
	return Float64(0.5641895835477563 * abs(Float64(2.0 * abs(x))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5641895835477563 * abs((2.0 * abs(x)));
end

Wolfram

code[x_] := N[(0.5641895835477563 * N[Abs[N[(2.0 * N[Abs[x], $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%

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

5. Applied rewrites 99.8%

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

6. Evaluated real constant 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \left|\color{blue}{2 \cdot \left|x\right|}\right| \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \left|2 \cdot \color{blue}{\left|x\right|}\right| \]

   2. lower-fabs.f64 67.7

      \[\leadsto 0.5641895835477563 \cdot \left|2 \cdot \left|x\right|\right| \]

9. Applied rewrites 67.7%

   \[\leadsto 0.5641895835477563 \cdot \left|\color{blue}{2 \cdot \left|x\right|}\right| \]
10. Add Preprocessing

Alternative 11: 67.7% accurate, 15.0× speedup

Math

\[\left|\left|x\right| \cdot 1.1283791670955126\right| \]

FPCore

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

C

double code(double x) {
	return fabs((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((abs(x) * 1.1283791670955126d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[Abs[x], $MachinePrecision] * 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 99.8%

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

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
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 67.2

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

6. Applied rewrites 67.2%

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

7. Evaluated real constant 67.5%

   \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}\right| \]
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. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 67.7

       \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

9. Applied rewrites 67.7%

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

Reproduce

herbie shell --seed 2025173 
(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)))))))