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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 17

Speedup: 1.0×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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]]]

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. Add Preprocessing

Alternative 2: 99.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (fabs
    (*
     (fabs x)
     (/ (fma (* x x) (fma (* x x) 0.2 0.6666666666666666) 2.0) (sqrt PI))))
   (fabs (* (/ 1.0 (sqrt PI)) (* (pow x 7.0) 0.047619047619047616)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = fabs((fabs(x) * (fma((x * x), fma((x * x), 0.2, 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (pow(x, 7.0) * 0.047619047619047616)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = abs(Float64(abs(x) * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.2, 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 93.8

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

   6. Applied rewrites 93.8%

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

   if 2.60000000000000009 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 36.3

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

   6. Applied rewrites 36.3%

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

Alternative 3: 99.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.2%

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

Alternative 4: 98.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(-0.047619047619047616 \cdot t\_0\right) \cdot t\_0\right) \cdot x\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x 2.6)
     (fabs
      (*
       (fabs x)
       (/ (fma (* x x) (fma (* x x) 0.2 0.6666666666666666) 2.0) (sqrt PI))))
     (/ (fabs (* (* (* -0.047619047619047616 t_0) t_0) x)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= 2.6) {
		tmp = fabs((fabs(x) * (fma((x * x), fma((x * x), 0.2, 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((-0.047619047619047616 * t_0) * t_0) * x)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = abs(Float64(abs(x) * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.2, 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(-0.047619047619047616 * t_0) * t_0) * x)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 2.6], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(-0.047619047619047616 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 93.8

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

   6. Applied rewrites 93.8%

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

   if 2.60000000000000009 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 36.3

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

   6. Applied rewrites 36.3%

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

   8. Applied rewrites 36.3%

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

Alternative 5: 98.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7} \cdot 0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (fabs
    (*
     (fabs x)
     (/ (fma (* x x) (fma (* x x) 0.2 0.6666666666666666) 2.0) (sqrt PI))))
   (fabs (/ (* (pow x 7.0) 0.047619047619047616) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = fabs((fabs(x) * (fma((x * x), fma((x * x), 0.2, 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((pow(x, 7.0) * 0.047619047619047616) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = abs(Float64(abs(x) * Float64(fma(Float64(x * x), fma(Float64(x * x), 0.2, 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(Float64((x ^ 7.0) * 0.047619047619047616) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 93.8

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

   6. Applied rewrites 93.8%

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

   if 2.60000000000000009 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 36.3

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

   8. Applied rewrites 36.3%

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

Alternative 6: 93.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.3%

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

Alternative 7: 93.8% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((fabs(x) * (fma((x * x), 0.6666666666666666, 2.0) / sqrt(((double) M_PI)))));
	} 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 = abs(Float64(abs(x) * Float64(fma(Float64(x * x), 0.6666666666666666, 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(Float64((x ^ 7.0) * 0.047619047619047616) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.6%

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

   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 99.7%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 36.3

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

   8. Applied rewrites 36.3%

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

Alternative 8: 93.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(-0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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 98.8%

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

Alternative 9: 93.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. associate-*l* N/A

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

   4. pow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.4

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

6. Applied rewrites 98.4%

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

Alternative 10: 92.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 93.3

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

6. Applied rewrites 93.3%

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

7. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 92.9

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

9. Applied rewrites 92.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

11. Applied rewrites 93.4%

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

Alternative 11: 89.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 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 98.8%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 93.3

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

6. Applied rewrites 93.3%

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

7. Taylor expanded in x around inf

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

   1. pow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 92.9

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

9. Applied rewrites 92.9%

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

11. Applied rewrites 92.9%

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

Alternative 12: 89.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 89.6%

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

Alternative 13: 89.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 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 98.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. mul-fabs N/A

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

   5. lower-fabs.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 89.2

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 89.2

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

8. Applied rewrites 89.2%

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

Alternative 14: 68.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot x\right) \cdot \frac{2 \cdot x}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (fabs (* (/ 2.0 (sqrt PI)) x))
   (sqrt (* (* 2.0 x) (/ (* 2.0 x) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * x));
	} else {
		tmp = sqrt(((2.0 * x) * ((2.0 * x) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = Math.sqrt(((2.0 * x) * ((2.0 * x) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * x))
	else:
		tmp = math.sqrt(((2.0 * x) * ((2.0 * x) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * x));
	else
		tmp = sqrt(Float64(Float64(2.0 * x) * Float64(Float64(2.0 * x) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = abs(((2.0 / sqrt(pi)) * x));
	else
		tmp = sqrt(((2.0 * x) * ((2.0 * x) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * N[(N[(2.0 * x), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   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 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. fabs-mul N/A

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

      6. fabs-fabs N/A

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

      7. mul-fabs N/A

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

      8. lower-fabs.f64 N/A

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

   8. Applied rewrites 68.3%

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

   if 0.5 < 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 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. fabs-mul N/A

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

      6. fabs-fabs N/A

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

      7. mul-fabs N/A

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

      8. lower-fabs.f64 N/A

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

   8. Applied rewrites 68.3%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

   10. Applied rewrites 53.3%

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

Alternative 15: 68.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 10^{-154}:\\ \;\;\;\;\left|t\_0 \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{x \cdot x} \cdot t\_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt PI))))
   (if (<= x 1e-154) (fabs (* t_0 x)) (fabs (* (sqrt (* x x)) t_0)))))

C

double code(double x) {
	double t_0 = 2.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 1e-154) {
		tmp = fabs((t_0 * x));
	} else {
		tmp = fabs((sqrt((x * x)) * t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 2.0 / Math.sqrt(Math.PI);
	double tmp;
	if (x <= 1e-154) {
		tmp = Math.abs((t_0 * x));
	} else {
		tmp = Math.abs((Math.sqrt((x * x)) * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 2.0 / math.sqrt(math.pi)
	tmp = 0
	if x <= 1e-154:
		tmp = math.fabs((t_0 * x))
	else:
		tmp = math.fabs((math.sqrt((x * x)) * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(2.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 1e-154)
		tmp = abs(Float64(t_0 * x));
	else
		tmp = abs(Float64(sqrt(Float64(x * x)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 2.0 / sqrt(pi);
	tmp = 0.0;
	if (x <= 1e-154)
		tmp = abs((t_0 * x));
	else
		tmp = abs((sqrt((x * x)) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e-154], N[Abs[N[(t$95$0 * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999997e-155

   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 99.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. *-commutative N/A

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

      5. fabs-mul N/A

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

      6. fabs-fabs N/A

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

      7. mul-fabs N/A

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

      8. lower-fabs.f64 N/A

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

   8. Applied rewrites 62.1%

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

   if 9.9999999999999997e-155 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.3%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 98.2

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

   8. Applied rewrites 98.2%

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

Alternative 16: 68.3% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.3%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. *-commutative N/A

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

   5. fabs-mul N/A

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

   6. fabs-fabs N/A

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

   7. mul-fabs N/A

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

   8. lower-fabs.f64 N/A

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

8. Applied rewrites 68.3%

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

Alternative 17: 67.9% 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 98.8%

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

4. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 93.3

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

6. Applied rewrites 93.3%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. mul-fabs N/A

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

   5. lower-fabs.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 93.3

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

8. Applied rewrites 93.3%

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

9. Taylor expanded in x around 0

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

    1. count-2-rev N/A

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

    2. lower-+.f64 67.9

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

11. Applied rewrites 67.9%

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

Reproduce

herbie shell --seed 2025130 
(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)))))))