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

Percentage Accurate: 99.8% → 99.8%

Time: 2.8s

Alternatives: 13

Speedup: 2.2×

• 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: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow-plus N/A

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

   14. pow3 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

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

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

   2. pow2 N/A

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

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

   5. unpow1 77.5

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

9. Applied rewrites 77.5%

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

10. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 99.8

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

12. Applied rewrites 99.8%

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

    1. lift-fma.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. distribute-lft-out N/A

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

14. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.4%

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

Alternative 3: 99.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fma((x * x), 0.6666666666666666, 2.0) * x));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / 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(fma(Float64(x * x), 0.6666666666666666, 2.0) * x)));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / 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[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $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%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2 N/A

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

      3. sqr-abs-rev N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 89.0%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

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

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. pow3 N/A

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

      13. pow2 N/A

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

      14. sqr-abs-rev N/A

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

      15. associate-*l* N/A

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

   6. Applied rewrites 36.6%

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

Alternative 4: 98.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow-plus N/A

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

   14. pow3 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

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

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

   2. pow2 N/A

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

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

   5. unpow1 77.5

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

9. Applied rewrites 77.5%

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

10. Taylor expanded in x around 0

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

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

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

    2. pow2 N/A

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

    3. sqrt-pow1 N/A

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

    4. metadata-eval N/A

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

    5. unpow1 99.8

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

12. Applied rewrites 99.8%

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

13. Taylor expanded in x around 0

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

15. Applied rewrites 99.0%

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

Alternative 5: 89.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.8%

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

Alternative 6: 89.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|0.2 \cdot \left(t\_0 \cdot t\_0\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x 1.85)
     (* (/ 1.0 (sqrt PI)) (fabs (* (fma (* x x) 0.6666666666666666 2.0) x)))
     (/ (fabs (* 0.2 (* t_0 t_0))) (sqrt PI)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= 1.85) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fma((x * x), 0.6666666666666666, 2.0) * x));
	} else {
		tmp = fabs((0.2 * (t_0 * t_0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(Float64(x * x), 0.6666666666666666, 2.0) * x)));
	else
		tmp = Float64(abs(Float64(0.2 * Float64(t_0 * t_0))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1.85], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(0.2 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2 N/A

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

      3. sqr-abs-rev N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 89.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 34.6%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\left|\left(\frac{\frac{2}{\frac{0}{1 - 1}}}{x} \cdot \frac{1}{5}\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right|}{\sqrt{\pi}} \]

      9. +-inverses N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\left(\frac{\frac{2}{\frac{x \cdot x - x \cdot x}{0}}}{x} \cdot \frac{1}{5}\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right|}{\sqrt{\pi}} \]

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. count-2-rev N/A

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

      14. associate-/l/ N/A

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

      15. metadata-eval N/A

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

      16. associate-/r* N/A

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

      17. pow2 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. metadata-eval N/A

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

      21. pow2 N/A

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

      22. associate-*l* N/A

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

   9. Applied rewrites 31.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lift-*.f64 N/A

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

       7. swap-sqr N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 31.8

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

   11. Applied rewrites 31.8%

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

Alternative 7: 89.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;t\_0 \cdot \left|\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.2\right) \cdot x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.3)
     (* t_0 (fabs (* (fma (* x x) 0.6666666666666666 2.0) x)))
     (fabs (* t_0 (* (* (* (* (* x x) x) x) 0.2) x))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.3) {
		tmp = t_0 * fabs((fma((x * x), 0.6666666666666666, 2.0) * x));
	} else {
		tmp = fabs((t_0 * (((((x * x) * x) * x) * 0.2) * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.3)
		tmp = Float64(t_0 * abs(Float64(fma(Float64(x * x), 0.6666666666666666, 2.0) * x)));
	else
		tmp = abs(Float64(t_0 * Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * 0.2) * x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.3], N[(t$95$0 * N[Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(t$95$0 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.2), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2 N/A

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

      3. sqr-abs-rev N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 89.0%

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

   if 2.2999999999999998 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.2\right) \cdot \color{blue}{x}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. pow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. sqr-abs-rev N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 89.0%

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

Alternative 9: 88.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x + x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2.6666666666666665 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.85)
   (* (/ 1.0 (sqrt PI)) (fabs (+ x x)))
   (/ (fabs (* 2.6666666666666665 (* (* x x) x))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 0.85) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x + x));
	} else {
		tmp = fabs((2.6666666666666665 * ((x * x) * x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.85) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x + x));
	} else {
		tmp = Math.abs((2.6666666666666665 * ((x * x) * x))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.85:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x + x))
	else:
		tmp = math.fabs((2.6666666666666665 * ((x * x) * x))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.85)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x + x)));
	else
		tmp = Float64(abs(Float64(2.6666666666666665 * Float64(Float64(x * x) * x))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.85)
		tmp = (1.0 / sqrt(pi)) * abs((x + x));
	else
		tmp = abs((2.6666666666666665 * ((x * x) * x))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.85], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(2.6666666666666665 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.849999999999999978

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 N/A

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

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

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

      9. pow2 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 67.8

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

   6. Applied rewrites 67.8%

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

   if 0.849999999999999978 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2 N/A

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

      3. sqr-abs-rev N/A

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

      4. sqr-abs-rev N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 89.0%

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

   8. Applied rewrites 26.9%

      \[\leadsto \color{blue}{\frac{\left|2.6666666666666665 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right|}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.8% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. pow2 N/A

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

   3. sqr-abs-rev N/A

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

   4. sqr-abs-rev N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 88.6%

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

Alternative 11: 67.8% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 N/A

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

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

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

   4. pow2 N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

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

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. metadata-eval N/A

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

   12. unpow1 67.8

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

6. Applied rewrites 67.8%

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

Alternative 12: 67.4% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 N/A

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

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

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

   4. pow2 N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

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

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. metadata-eval N/A

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

   12. unpow1 67.4

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

6. Applied rewrites 67.4%

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

Alternative 13: 4.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|0.2\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fabs 0.2) (sqrt PI)))

C

double code(double x) {
	return fabs(0.2) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return Math.abs(0.2) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs(0.2) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(abs(0.2) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = abs(0.2) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[Abs[0.2], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 34.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 4.7%

   \[\leadsto \frac{\left|0.2\right|}{\sqrt{\pi}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(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)))))))