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

Percentage Accurate: 99.8% → 99.9%

Time: 5.5s

Alternatives: 13

Speedup: 1.7×

• 29.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 3: 99.4% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (fma
     (* (/ (fabs x) (sqrt PI)) (* x x))
     0.6666666666666666
     (* (fabs x) (/ 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(fma(((fabs(x) / sqrt(((double) M_PI))) * (x * x)), 0.6666666666666666, (fabs(x) * (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(fma(Float64(Float64(abs(x) / sqrt(pi)) * Float64(x * x)), 0.6666666666666666, Float64(abs(x) * Float64(2.0 / sqrt(pi)))));
	else
		tmp = abs(Float64((Float64(-x) ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-PI.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 88.7

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 88.7

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

   8. Applied rewrites 89.1%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 37.8

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

   6. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.8

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lift-fabs.f64 N/A

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

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

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

      11. pow1/2 N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow2 N/A

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

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

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

      17. lift-fabs.f64 N/A

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

      18. lift-pow.f64 37.8

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

   8. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fabs.f64 N/A

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

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

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

      8. sqr-neg-rev N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-/.f64 37.8

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

   10. Applied rewrites 37.8%

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

Alternative 4: 99.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $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 99.9%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.4%

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

Alternative 5: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(0.047619047619047616 * x), $MachinePrecision] * x + 0.2), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision] * N[(x / 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 99.9%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 6: 98.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\left(-x\right)}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs (* (pow (- x) 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((pow(-x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((Math.pow(-x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((math.pow(-x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64((Float64(-x) ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs(((-x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[(-x), 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $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 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.5

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

   6. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.0

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

   8. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.0

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

   10. Applied rewrites 67.0%

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

   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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 37.8

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

   6. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.8

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lift-fabs.f64 N/A

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

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

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

      11. pow1/2 N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. sqrt-pow2 N/A

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

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

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

      17. lift-fabs.f64 N/A

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

      18. lift-pow.f64 37.8

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

   8. Applied rewrites 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fabs.f64 N/A

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

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

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

      8. sqr-neg-rev N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-/.f64 37.8

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

   10. Applied rewrites 37.8%

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

Alternative 7: 89.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * 0.047619047619047616)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * 0.047619047619047616)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs(((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $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 < 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 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.5

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

   6. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.0

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

   8. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.0

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

   10. Applied rewrites 67.0%

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

   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 99.9%

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 37.8

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

   8. Applied rewrites 37.8%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{0.047619047619047616 \cdot {x}^{7}}\right| \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 37.8

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

   10. Applied rewrites 37.8%

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

Alternative 8: 88.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, Float64(2.0 * abs(x)))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 99.0

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

6. Applied rewrites 99.0%

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

Alternative 9: 83.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-PI.f64 98.6

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

6. Applied rewrites 98.6%

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

Alternative 10: 67.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 88.7

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

6. Applied rewrites 88.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f64 N/A

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

8. Applied rewrites 88.7%

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

Alternative 11: 67.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (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))))))
        5e-37)
     (fabs (* (/ 2.0 (sqrt PI)) (- x)))
     (fabs (* 2.0 (sqrt (/ (* x x) PI)))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (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)))))) <= 5e-37) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (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)))))) <= 5e-37) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if 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)))))) <= 5e-37:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (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)))))) <= 5e-37)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (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)))))) <= 5e-37)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((2.0 * sqrt(((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 5e-37], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 4.9999999999999997e-37

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.5

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

   6. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.0

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

   8. Applied rewrites 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.0

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.0

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

   10. Applied rewrites 67.0%

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

   if 4.9999999999999997e-37 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.5

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

   6. Applied rewrites 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 53.8

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

   8. Applied rewrites 53.8%

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

Alternative 12: 67.0% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 66.5

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

6. Applied rewrites 66.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 67.0

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

8. Applied rewrites 67.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 67.0

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

   4. lift-fabs.f64 N/A

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

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

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

   6. sqr-neg-rev N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. rem-square-sqrt 67.0

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

10. Applied rewrites 67.0%

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

Alternative 13: 66.5% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 2.0), $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 99.9%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025162 
(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)))))))