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

Percentage Accurate: 99.8% → 99.9%

Time: 5.0s

Alternatives: 12

Speedup: 2.0×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(fma(Float64(-0.2 - Float64(Float64(0.047619047619047616 * x) * x)), Float64(Float64(Float64(Float64(Float64(-x) * x) * x) * x) * x), 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[(-0.2 - N[(N[(0.047619047619047616 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[((-x) * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.9%

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

Alternative 4: 99.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), 0.2, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (*
    (/ 1.0 (sqrt PI))
    (fabs
     (fma
      (* (* (fabs x) x) (* (* x x) x))
      0.2
      (* (fma 0.6666666666666666 (* x x) 2.0) (fabs x)))))
   (fabs (* 0.047619047619047616 (/ (* (pow x 6.0) (fabs x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs(fma(((fabs(x) * x) * ((x * x) * x)), 0.2, (fma(0.6666666666666666, (x * x), 2.0) * fabs(x))));
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(x, 6.0) * fabs(x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(fma(Float64(Float64(abs(x) * x) * Float64(Float64(x * x) * x)), 0.2, Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * abs(x)))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * 0.2 + N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.8%

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

   if 2.60000000000000009 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 36.3

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

   6. Applied rewrites 36.3%

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

Alternative 5: 93.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\left(-x\right) \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (* (/ 1.0 (sqrt PI)) (fabs (* (- x) (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs (* 0.047619047619047616 (/ (* (pow x 6.0) (fabs x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((-x * fma(0.6666666666666666, (x * x), 2.0)));
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(x, 6.0) * fabs(x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(Float64(-x) * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[((-x) * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $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%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 89.2

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

   6. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 89.2

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

   8. Applied rewrites 89.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fabs-mul N/A

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

   10. Applied rewrites 89.2%

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

   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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 36.3

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

   6. Applied rewrites 36.3%

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

Alternative 6: 89.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[((-x) * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(0.047619047619047616 * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 89.2

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

   6. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 89.2

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

   8. Applied rewrites 89.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fabs-mul N/A

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

   10. Applied rewrites 89.2%

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

   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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 36.3

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

   6. Applied rewrites 36.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)}\right| \]

   8. Applied rewrites 36.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot 0.047619047619047616\right) \cdot \left(\left(\left(\left|x\right| \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{x}\right)\right| \]

   10. Applied rewrites 36.3%

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

Alternative 7: 89.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(-0.2 - N[(N[(0.047619047619047616 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[((-x) * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * 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 99.8%

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.4%

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

Alternative 8: 89.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 89.2

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

6. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 89.2

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

8. Applied rewrites 89.2%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fabs-mul N/A

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

10. Applied rewrites 89.2%

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

Alternative 9: 88.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[((-x) * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $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 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 89.2

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

6. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 89.2

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

8. Applied rewrites 89.2%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

10. Applied rewrites 88.7%

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

Alternative 10: 68.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-24)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs (* 2.0 (sqrt (/ (* x x) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-24) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-24) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-24:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-24)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-24)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((2.0 * sqrt(((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-24], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999985e-24

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 68.2

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.2

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 68.2

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

   10. Applied rewrites 68.2%

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

   if 1.99999999999999985e-24 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 53.9

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

   8. Applied rewrites 53.9%

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

Alternative 11: 68.2% accurate, 8.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs(((2.0 / sqrt(((double) M_PI))) * -x));
}

Java

public static double code(double x) {
	return Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
}

Python

def code(x):
	return math.fabs(((2.0 / math.sqrt(math.pi)) * -x))

Julia

function code(x)
	return abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = abs(((2.0 / sqrt(pi)) * -x));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 67.8

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

6. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 68.2

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

8. Applied rewrites 68.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 68.2

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

   4. lift-fabs.f64 N/A

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

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

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

   6. sqr-neg-rev N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. rem-square-sqrt 68.2

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

10. Applied rewrites 68.2%

    \[\leadsto \left|\frac{2}{\sqrt{\pi}} \cdot \color{blue}{\left(-x\right)}\right| \]
11. Add Preprocessing

Alternative 12: 67.8% accurate, 8.7× speedup

Math

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

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 abs(Float64(Float64(Float64(-x) - x) / sqrt(pi)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 67.8

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

6. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 68.2

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

8. Applied rewrites 68.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-/.f64 67.8

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

10. Applied rewrites 67.8%

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

Reproduce

herbie shell --seed 2025149 
(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)))))))