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

Percentage Accurate: 99.8% → 99.8%

Time: 4.3s

Alternatives: 9

Speedup: 2.0×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 98.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(\left(0.047619047619047616 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
   (fabs
    (*
     (* (* (* (* 0.047619047619047616 (* x x)) x) (* x x)) x)
     (/ (fabs x) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((((0.047619047619047616 * (x * x)) * x) * (x * x)) * x) * (fabs(x) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((((((0.047619047619047616 * (x * x)) * x) * (x * x)) * x) * (Math.abs(x) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((((((0.047619047619047616 * (x * x)) * x) * (x * x)) * x) * (math.fabs(x) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(0.047619047619047616 * Float64(x * x)) * x) * Float64(x * x)) * x) * Float64(abs(x) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs((((((0.047619047619047616 * (x * x)) * x) * (x * x)) * x) * (abs(x) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-add-rev N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 37.0

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

   6. Applied rewrites 37.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.1

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

   10. Applied rewrites 37.1%

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

Alternative 3: 83.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 98.9%

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

Alternative 4: 67.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(0.047619047619047616 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x 1.9)
     (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
     (fabs (* (* 0.047619047619047616 (* t_0 t_0)) (/ (fabs x) (sqrt PI)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= 1.9) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((0.047619047619047616 * (t_0 * t_0)) * (fabs(x) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((0.047619047619047616 * (t_0 * t_0)) * (Math.abs(x) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * x
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(((0.047619047619047616 * (t_0 * t_0)) * (math.fabs(x) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 * Float64(t_0 * t_0)) * Float64(abs(x) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * x;
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs(((0.047619047619047616 * (t_0 * t_0)) * (abs(x) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1.9], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-add-rev N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 37.0

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

   6. Applied rewrites 37.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 37.1%

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

Alternative 5: 67.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} 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 <= 1.9) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} 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 <= 1.9:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	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 <= 1.9)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $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 < 1.8999999999999999

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-add-rev N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 37.0

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

   6. Applied rewrites 37.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   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. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\left|{\left(x \cdot x\right)}^{3.5} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|} \]

   9. Taylor expanded in x around 0

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

       1. lower-pow.f64 37.1

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

   11. Applied rewrites 37.1%

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

Alternative 6: 67.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{\sqrt{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-25)
   (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
   (fabs (* 2.0 (sqrt (/ (sqrt (* (* (* x x) x) x)) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-25) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((2.0 * sqrt((sqrt((((x * x) * x) * x)) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-25) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((Math.sqrt((((x * x) * x) * x)) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-25:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((2.0 * math.sqrt((math.sqrt((((x * x) * x) * x)) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-25)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(sqrt(Float64(Float64(Float64(x * x) * x) * x)) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-25)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs((2.0 * sqrt((sqrt((((x * x) * x) * x)) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-25], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000008e-25

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-add-rev N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 2.00000000000000008e-25 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-sqrt.f64 45.0

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

   10. Applied rewrites 45.0%

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

Alternative 7: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

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))))))
        2e-25)
     (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
     (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)))))) <= 2e-25) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} 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)))))) <= 2e-25) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} 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)))))) <= 2e-25:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	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)))))) <= 2e-25)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	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)))))) <= 2e-25)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	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], 2e-25], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $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)))))) < 2.00000000000000008e-25

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. div-add-rev N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 2.00000000000000008e-25 < (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 99.8%

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

   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 67.1

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

   6. Applied rewrites 67.1%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   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 52.7

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

   8. Applied rewrites 52.7%

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

Alternative 8: 67.5% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

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 67.1

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

6. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. lift-/.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. div-add-rev N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 67.5

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

8. Applied rewrites 67.5%

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

Alternative 9: 67.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

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 67.1

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

6. Applied rewrites 67.1%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))