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

Percentage Accurate: 99.8% → 99.9%

Time: 5.3s

Alternatives: 14

Speedup: 2.6×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

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

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

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: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \mathsf{fma}\left({x}^{6}, 0.047619047619047616, 2\right)\right) \cdot x\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs
   (*
    (fma
     (fma (* x x) 0.2 0.6666666666666666)
     (* x x)
     (fma (pow x 6.0) 0.047619047619047616 2.0))
    x))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((fma(fma((x * x), 0.2, 0.6666666666666666), (x * x), fma(pow(x, 6.0), 0.047619047619047616, 2.0)) * x));
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(fma(Float64(x * x), 0.2, 0.6666666666666666), Float64(x * x), fma((x ^ 6.0), 0.047619047619047616, 2.0)) * x)))
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * 0.047619047619047616 + 2.0), $MachinePrecision]), $MachinePrecision] * x), $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%

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

8. Applied rewrites 99.9%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21}\right) + 2\right) \cdot x\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21}\right) + 2\right) \cdot x\right| \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right) \cdot \left(x \cdot x\right) + {x}^{6} \cdot \frac{1}{21}\right) + 2\right) \cdot x\right| \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right) \cdot \left(x \cdot x\right) + {x}^{6} \cdot \frac{1}{21}\right) + 2\right) \cdot x\right| \]

   5. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{2}{3}\right) \cdot \left(x \cdot x\right) + {x}^{6} \cdot \frac{1}{21}\right) + 2\right) \cdot x\right| \]

   6. associate-+l+ N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{2}{3}\right) \cdot \left(x \cdot x\right) + \left({x}^{6} \cdot \frac{1}{21} + 2\right)\right) \cdot x\right| \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{2}{3}, x \cdot x, {x}^{6} \cdot \frac{1}{21} + 2\right) \cdot x\right| \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21} + 2\right) \cdot x\right| \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21} + 2\right) \cdot x\right| \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21} + 2\right) \cdot x\right| \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21} + 2\right) \cdot x\right| \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, {x}^{6} \cdot \frac{1}{21} + 2\right) \cdot x\right| \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, \mathsf{fma}\left({x}^{6}, \frac{1}{21}, 2\right)\right) \cdot x\right| \]

   14. lift-pow.f64 99.9

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \mathsf{fma}\left({x}^{6}, 0.047619047619047616, 2\right)\right) \cdot x\right| \]

10. Applied rewrites 99.9%

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, \mathsf{fma}\left({x}^{6}, 0.047619047619047616, 2\right)\right) \cdot x\right| \]
11. Add Preprocessing

Alternative 2: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   4. pow2 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

   8. metadata-eval N/A

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

   9. pow-plus N/A

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

   10. lower-*.f64 N/A

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

   11. pow3 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

8. Applied rewrites 99.9%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right) \cdot x\right| \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. pow2 N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. +-commutative N/A

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

    6. pow2 N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. +-commutative N/A

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

    10. pow2 N/A

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

    11. lower-fma.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. lift-*.f64 N/A

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

    14. lift-*.f64 99.8

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

11. Applied rewrites 99.8%

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

Alternative 4: 98.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left({x}^{6} \cdot 0.047619047619047616\right) \cdot \left|x\right|\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.65)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* (fma (fma (* x x) 0.2 0.6666666666666666) (* x x) 2.0) x)))
   (/ (fabs (* (* (pow x 6.0) 0.047619047619047616) (fabs x))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.65) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fma(fma((x * x), 0.2, 0.6666666666666666), (x * x), 2.0) * x));
	} else {
		tmp = fabs(((pow(x, 6.0) * 0.047619047619047616) * fabs(x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.65)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(fma(Float64(x * x), 0.2, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(abs(Float64(Float64((x ^ 6.0) * 0.047619047619047616) * abs(x))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.65], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[Power[x, 6.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999991

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x\right| \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. *-commutative N/A

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

       4. +-commutative N/A

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

       5. pow2 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right| \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right| \]

       10. lift-*.f64 93.1

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

   11. Applied rewrites 93.1%

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

   if 2.64999999999999991 < 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%

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

   4. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-fabs.f64 35.8

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

   6. Applied rewrites 35.8%

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

Alternative 5: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;t\_0 \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|{x}^{7} \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.65)
     (*
      t_0
      (fabs (* (fma (fma (* x x) 0.2 0.6666666666666666) (* x x) 2.0) x)))
     (* t_0 (fabs (* (pow x 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.65) {
		tmp = t_0 * fabs((fma(fma((x * x), 0.2, 0.6666666666666666), (x * x), 2.0) * x));
	} else {
		tmp = t_0 * fabs((pow(x, 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.65)
		tmp = Float64(t_0 * abs(Float64(fma(fma(Float64(x * x), 0.2, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(t_0 * abs(Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.65], N[(t$95$0 * N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999991

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right) \cdot x\right| \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. *-commutative N/A

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

       4. +-commutative N/A

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

       5. pow2 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right| \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{5}, \frac{2}{3}\right), x \cdot x, 2\right) \cdot x\right| \]

       10. lift-*.f64 93.1

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

   11. Applied rewrites 93.1%

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

   if 2.64999999999999991 < 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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\frac{1}{21} \cdot {x}^{7}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 35.8

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot 0.047619047619047616\right| \]

   11. Applied rewrites 35.8%

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot 0.047619047619047616\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;t\_0 \cdot \left|\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|{x}^{7} \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.2)
     (* t_0 (fabs (* (fma 0.6666666666666666 (* x x) 2.0) x)))
     (* t_0 (fabs (* (pow x 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.2) {
		tmp = t_0 * fabs((fma(0.6666666666666666, (x * x), 2.0) * x));
	} else {
		tmp = t_0 * fabs((pow(x, 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(t_0 * abs(Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(t_0 * abs(Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[(t$95$0 * N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + \frac{2}{3} \cdot {x}^{2}\right) \cdot x\right| \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right) \cdot x\right| \]

       4. lift-*.f64 88.9

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

   11. Applied rewrites 88.9%

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

   if 2.2000000000000002 < 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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\frac{1}{21} \cdot {x}^{7}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 35.8

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot 0.047619047619047616\right| \]

   11. Applied rewrites 35.8%

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot 0.047619047619047616\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return abs(Float64(fma((abs(x) ^ 7.0), -0.047619047619047616, Float64(-2.0 * abs(x))) * Float64(1.0 / sqrt(pi))))
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] * N[(1.0 / N[Sqrt[Pi], $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 73.8%

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

4. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-fabs.f64 N/A

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

   11. inv-pow N/A

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

   12. sqrt-pow1 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

6. Applied rewrites 98.8%

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

Alternative 8: 88.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left|{\left(\left|x\right|\right)}^{7} \cdot -0.047619047619047616 - \left(x + x\right)\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fabs (- (* (pow (fabs x) 7.0) -0.047619047619047616) (+ x x))) (sqrt PI)))

C

double code(double x) {
	return fabs(((pow(fabs(x), 7.0) * -0.047619047619047616) - (x + x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return Math.abs(((Math.pow(Math.abs(x), 7.0) * -0.047619047619047616) - (x + x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs(((math.pow(math.fabs(x), 7.0) * -0.047619047619047616) - (x + x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(abs(Float64(Float64((abs(x) ^ 7.0) * -0.047619047619047616) - Float64(x + x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = abs((((abs(x) ^ 7.0) * -0.047619047619047616) - (x + x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * -0.047619047619047616), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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 73.8%

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

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. fp-cancel-sub-sign-inv N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lift-fabs.f64 98.4

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

6. Applied rewrites 98.4%

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

8. Applied rewrites 98.4%

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

10. Applied rewrites 98.4%

    \[\leadsto \color{blue}{\frac{\left|{\left(\left|x\right|\right)}^{7} \cdot -0.047619047619047616 - \left(x + x\right)\right|}{\sqrt{\pi}}} \]
11. Add Preprocessing

Alternative 9: 88.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.25)
   (* (/ 1.0 (sqrt PI)) (fabs (* (fma 0.6666666666666666 (* x x) 2.0) x)))
   (fabs (/ (* -0.2 (* (* (* x x) (* x x)) x)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fma(0.6666666666666666, (x * x), 2.0) * x));
	} else {
		tmp = fabs(((-0.2 * (((x * x) * (x * x)) * x)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * x)));
	else
		tmp = abs(Float64(Float64(-0.2 * Float64(Float64(Float64(x * x) * Float64(x * x)) * x)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.25], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(-0.2 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + \frac{2}{3} \cdot {x}^{2}\right) \cdot x\right| \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right) \cdot x\right| \]

       4. lift-*.f64 88.9

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

   11. Applied rewrites 88.9%

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

   if 2.25 < 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 73.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \color{blue}{{x}^{5}}}{\sqrt{\pi}}\right| \]

      2. metadata-eval N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot {x}^{\left(3 + \color{blue}{2}\right)}}{\sqrt{\pi}}\right| \]

      3. pow-prod-up N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left({x}^{3} \cdot \color{blue}{{x}^{2}}\right)}{\sqrt{\pi}}\right| \]

      4. pow2 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left({x}^{3} \cdot \left(x \cdot \color{blue}{x}\right)\right)}{\sqrt{\pi}}\right| \]

      5. associate-*l* N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left({x}^{3} \cdot x\right) \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left({x}^{3} \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      7. pow3 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      8. lift-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      10. lift-*.f64 29.7

          \[\leadsto \left|\frac{-0.2 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 29.7%

      \[\leadsto \left|\frac{\color{blue}{-0.2 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      4. pow3 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left({x}^{3} \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      5. unpow3 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      6. pow2 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left({x}^{2} \cdot x\right) \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      7. associate-*l* N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left({x}^{2} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      8. pow2 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      9. lower-*.f64 N/A

         \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      10. pow2 N/A

          \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      11. lift-*.f64 N/A

          \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      12. pow2 N/A

          \[\leadsto \left|\frac{\frac{-1}{5} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

      13. lift-*.f64 29.7

          \[\leadsto \left|\frac{-0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\sqrt{\pi}}\right| \]

   8. Applied rewrites 29.7%

      \[\leadsto \left|\frac{-0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right| \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

8. Applied rewrites 99.9%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + \frac{2}{3} \cdot {x}^{2}\right) \cdot x\right| \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. pow2 N/A

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

    3. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right) \cdot x\right| \]

    4. lift-*.f64 88.9

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

11. Applied rewrites 88.9%

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

Alternative 11: 88.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left|x\right|\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fabs (* (fma (* x x) 0.6666666666666666 2.0) (fabs x))) (sqrt PI)))

C

double code(double x) {
	return fabs((fma((x * x), 0.6666666666666666, 2.0) * fabs(x))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(abs(Float64(fma(Float64(x * x), 0.6666666666666666, 2.0) * abs(x))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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%

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

4. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right) \cdot \color{blue}{{x}^{6}}\right|}{\sqrt{\pi}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right) \cdot \color{blue}{{x}^{6}}\right|}{\sqrt{\pi}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}} + \frac{1}{21} \cdot \left|x\right|\right) \cdot {\color{blue}{x}}^{6}\right|}{\sqrt{\pi}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{\left|x\right|}{{x}^{2}} \cdot \frac{1}{5} + \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{{x}^{2}}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {\color{blue}{x}}^{6}\right|}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   8. lift-fabs.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   12. lift-fabs.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   13. lower-pow.f64 33.8

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, \left|x\right| \cdot 0.047619047619047616\right) \cdot {x}^{\color{blue}{6}}\right|}{\sqrt{\pi}} \]

6. Applied rewrites 33.8%

   \[\leadsto \frac{\left|\color{blue}{\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, \left|x\right| \cdot 0.047619047619047616\right) \cdot {x}^{6}}\right|}{\sqrt{\pi}} \]

7. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. distribute-rgt-in N/A

      \[\leadsto \frac{\left|\left|x\right| \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)}\right|}{\sqrt{\pi}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left|\left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) \cdot \color{blue}{\left|x\right|}\right|}{\sqrt{\pi}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) \cdot \color{blue}{\left|x\right|}\right|}{\sqrt{\pi}} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \left|\color{blue}{x}\right|\right|}{\sqrt{\pi}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \left|x\right|\right|}{\sqrt{\pi}} \]

   9. lift-fabs.f64 88.4

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

9. Applied rewrites 88.4%

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

Alternative 12: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \frac{1}{\sqrt{\pi}}\\ t_2 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|t\_1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left(t\_2 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 5 \cdot 10^{-13}:\\ \;\;\;\;t\_1 \cdot \left|x + x\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(x + x\right) \cdot \left(x + x\right)}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (/ 1.0 (sqrt PI)))
        (t_2 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          t_1
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_2))
           (* (/ 1.0 21.0) (* (* t_2 (fabs x)) (fabs x))))))
        5e-13)
     (* t_1 (fabs (+ x x)))
     (sqrt (/ (* (+ x x) (+ x x)) PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = 1.0 / sqrt(((double) M_PI));
	double t_2 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs((t_1 * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * fabs(x)) * fabs(x)))))) <= 5e-13) {
		tmp = t_1 * fabs((x + x));
	} else {
		tmp = sqrt((((x + x) * (x + x)) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = 1.0 / Math.sqrt(Math.PI);
	double t_2 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs((t_1 * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * Math.abs(x)) * Math.abs(x)))))) <= 5e-13) {
		tmp = t_1 * Math.abs((x + x));
	} else {
		tmp = Math.sqrt((((x + x) * (x + x)) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = 1.0 / math.sqrt(math.pi)
	t_2 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs((t_1 * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * math.fabs(x)) * math.fabs(x)))))) <= 5e-13:
		tmp = t_1 * math.fabs((x + x))
	else:
		tmp = math.sqrt((((x + x) * (x + x)) / math.pi))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(1.0 / sqrt(pi))
	t_2 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(t_1 * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_2 * abs(x)) * abs(x)))))) <= 5e-13)
		tmp = Float64(t_1 * abs(Float64(x + x)));
	else
		tmp = sqrt(Float64(Float64(Float64(x + x) * Float64(x + x)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = 1.0 / sqrt(pi);
	t_2 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs((t_1 * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * abs(x)) * abs(x)))))) <= 5e-13)
		tmp = t_1 * abs((x + x));
	else
		tmp = sqrt((((x + x) * (x + x)) / pi));
	end
	tmp_2 = tmp;
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[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$1 * 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$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-13], N[(t$95$1 * N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(x + x), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 4.9999999999999999e-13

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|2 \cdot x\right| \]
   10. Step-by-step derivation

       1. count-2-rev N/A

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \]

       2. lower-+.f64 68.8

          \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \]

   11. Applied rewrites 68.8%

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \]

   if 4.9999999999999999e-13 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 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%

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

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right) \cdot \color{blue}{{x}^{6}}\right|}{\sqrt{\pi}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right) \cdot \color{blue}{{x}^{6}}\right|}{\sqrt{\pi}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|\left(\frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}} + \frac{1}{21} \cdot \left|x\right|\right) \cdot {\color{blue}{x}}^{6}\right|}{\sqrt{\pi}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|\left(\frac{\left|x\right|}{{x}^{2}} \cdot \frac{1}{5} + \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{{x}^{2}}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {\color{blue}{x}}^{6}\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      8. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      12. lift-fabs.f64 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

      13. lower-pow.f64 33.8

          \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, \left|x\right| \cdot 0.047619047619047616\right) \cdot {x}^{\color{blue}{6}}\right|}{\sqrt{\pi}} \]

   6. Applied rewrites 33.8%

      \[\leadsto \frac{\left|\color{blue}{\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, \left|x\right| \cdot 0.047619047619047616\right) \cdot {x}^{6}}\right|}{\sqrt{\pi}} \]

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-fabs.f64 68.4

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

   9. Applied rewrites 68.4%

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

       1. lift-/.f64 N/A

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

       2. lift-fabs.f64 N/A

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

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

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

       4. lift-PI.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

   11. Applied rewrites 52.4%

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

Alternative 13: 68.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 (sqrt PI)) (fabs (+ x x))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((x + x));
}

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(Math.PI)) * Math.abs((x + x));
}

Python

def code(x):
	return (1.0 / math.sqrt(math.pi)) * math.fabs((x + x))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x + x)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(pi)) * abs((x + x));
end

Wolfram

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

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left|2 + \left(\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot {x}^{4} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right|\right)} \]

8. Applied rewrites 99.9%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, {x}^{6} \cdot 0.047619047619047616\right) + 2\right) \cdot x\right|} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|2 \cdot x\right| \]
10. Step-by-step derivation

    1. count-2-rev N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \]

    2. lower-+.f64 68.8

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \]

11. Applied rewrites 68.8%

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|x + x\right| \]
12. Add Preprocessing

Alternative 14: 68.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left|x + x\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fabs (+ x x)) (sqrt PI)))

C

double code(double x) {
	return fabs((x + x)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return Math.abs((x + x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs((x + x)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(abs(Float64(x + x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = abs((x + x)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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%

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

4. Taylor expanded in x around inf

   \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right) \cdot \color{blue}{{x}^{6}}\right|}{\sqrt{\pi}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right) \cdot \color{blue}{{x}^{6}}\right|}{\sqrt{\pi}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}} + \frac{1}{21} \cdot \left|x\right|\right) \cdot {\color{blue}{x}}^{6}\right|}{\sqrt{\pi}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left|\left(\frac{\left|x\right|}{{x}^{2}} \cdot \frac{1}{5} + \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{{x}^{2}}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {\color{blue}{x}}^{6}\right|}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   8. lift-fabs.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   12. lift-fabs.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \left|x\right| \cdot \frac{1}{21}\right) \cdot {x}^{6}\right|}{\sqrt{\pi}} \]

   13. lower-pow.f64 33.8

       \[\leadsto \frac{\left|\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, \left|x\right| \cdot 0.047619047619047616\right) \cdot {x}^{\color{blue}{6}}\right|}{\sqrt{\pi}} \]

6. Applied rewrites 33.8%

   \[\leadsto \frac{\left|\color{blue}{\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, \left|x\right| \cdot 0.047619047619047616\right) \cdot {x}^{6}}\right|}{\sqrt{\pi}} \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-fabs.f64 68.4

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

9. Applied rewrites 68.4%

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

    1. lift-*.f64 N/A

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

    2. lift-fabs.f64 N/A

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

    3. *-commutative N/A

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

    4. count-2-rev N/A

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

    5. flip3-+ N/A

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

11. Applied rewrites 68.4%

    \[\leadsto \frac{\left|\color{blue}{x + x}\right|}{\sqrt{\pi}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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)))))))