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

Percentage Accurate: 99.8% → 99.8%

Time: 5.3s

Alternatives: 11

Speedup: 2.0×

• 70.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.1× speedup

Math

\[\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 0.047619047619047616, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 2\right)\right)}{1.772453850905516}\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (/
    (fma
     (* (* (* (* (* x x) x) x) x) x)
     0.047619047619047616
     (fma (* x x) (fma (* 0.2 x) x 0.6666666666666666) 2.0))
    1.772453850905516))))

C

double code(double x) {
	return fabs(x) * fabs((fma((((((x * x) * x) * x) * x) * x), 0.047619047619047616, fma((x * x), fma((0.2 * x), x, 0.6666666666666666), 2.0)) / 1.772453850905516));
}

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(fma(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x), 0.047619047619047616, fma(Float64(x * x), fma(Float64(0.2 * x), x, 0.6666666666666666), 2.0)) / 1.772453850905516)))
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $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%

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8%

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

7. Applied rewrites 99.8%

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

8. Evaluated real constant 99.8%

   \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 0.047619047619047616, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 2\right)\right)}{\color{blue}{1.772453850905516}}\right| \]
9. Add Preprocessing

Alternative 3: 99.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 98.6%

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

8. Applied rewrites 99.1%

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

Alternative 4: 99.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \left|\frac{2 + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{2}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{21 \cdot \frac{\sqrt{\pi}}{{\left(\left|x\right|\right)}^{6} \cdot t\_0}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 3.1e-7)
     (*
      t_0
      (fabs (/ (+ 2.0 (* 0.6666666666666666 (pow (fabs x) 2.0))) (sqrt PI))))
     (fabs (/ 1.0 (* 21.0 (/ (sqrt PI) (* (pow (fabs x) 6.0) t_0))))))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 3.1e-7) {
		tmp = t_0 * fabs(((2.0 + (0.6666666666666666 * pow(fabs(x), 2.0))) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs((1.0 / (21.0 * (sqrt(((double) M_PI)) / (pow(fabs(x), 6.0) * t_0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 3.1e-7) {
		tmp = t_0 * Math.abs(((2.0 + (0.6666666666666666 * Math.pow(Math.abs(x), 2.0))) / Math.sqrt(Math.PI)));
	} else {
		tmp = Math.abs((1.0 / (21.0 * (Math.sqrt(Math.PI) / (Math.pow(Math.abs(x), 6.0) * t_0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 3.1e-7:
		tmp = t_0 * math.fabs(((2.0 + (0.6666666666666666 * math.pow(math.fabs(x), 2.0))) / math.sqrt(math.pi)))
	else:
		tmp = math.fabs((1.0 / (21.0 * (math.sqrt(math.pi) / (math.pow(math.fabs(x), 6.0) * t_0)))))
	return tmp

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 3.1e-7)
		tmp = Float64(t_0 * abs(Float64(Float64(2.0 + Float64(0.6666666666666666 * (abs(x) ^ 2.0))) / sqrt(pi))));
	else
		tmp = abs(Float64(1.0 / Float64(21.0 * Float64(sqrt(pi) / Float64((abs(x) ^ 6.0) * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(abs(x));
	tmp = 0.0;
	if (abs(x) <= 3.1e-7)
		tmp = t_0 * abs(((2.0 + (0.6666666666666666 * (abs(x) ^ 2.0))) / sqrt(pi)));
	else
		tmp = abs((1.0 / (21.0 * (sqrt(pi) / ((abs(x) ^ 6.0) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 3.1e-7], N[(t$95$0 * N[Abs[N[(N[(2.0 + N[(0.6666666666666666 * N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(1.0 / N[(21.0 * N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-7

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 89.2%

         \[\leadsto \left|x\right| \cdot \left|\frac{2 + 0.6666666666666666 \cdot {x}^{\color{blue}{2}}}{\sqrt{\pi}}\right| \]

   8. Applied rewrites 89.2%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{2 + 0.6666666666666666 \cdot {x}^{2}}}{\sqrt{\pi}}\right| \]

   if 3.1e-7 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 37.5%

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

   6. Applied rewrites 37.5%

      \[\leadsto \left|\frac{1}{\color{blue}{21 \cdot \frac{\sqrt{\pi}}{{x}^{6} \cdot \left|x\right|}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;\left|\left|x\right|\right| \cdot \left|\frac{2 + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{2}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left|{\left(\left|x\right|\right)}^{7}\right| \cdot 0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 3.1e-7)
   (*
    (fabs (fabs x))
    (fabs (/ (+ 2.0 (* 0.6666666666666666 (pow (fabs x) 2.0))) (sqrt PI))))
   (fabs (/ (* (fabs (pow (fabs x) 7.0)) 0.047619047619047616) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 3.1e-7) {
		tmp = fabs(fabs(x)) * fabs(((2.0 + (0.6666666666666666 * pow(fabs(x), 2.0))) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs(((fabs(pow(fabs(x), 7.0)) * 0.047619047619047616) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 3.1e-7) {
		tmp = Math.abs(Math.abs(x)) * Math.abs(((2.0 + (0.6666666666666666 * Math.pow(Math.abs(x), 2.0))) / Math.sqrt(Math.PI)));
	} else {
		tmp = Math.abs(((Math.abs(Math.pow(Math.abs(x), 7.0)) * 0.047619047619047616) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 3.1e-7:
		tmp = math.fabs(math.fabs(x)) * math.fabs(((2.0 + (0.6666666666666666 * math.pow(math.fabs(x), 2.0))) / math.sqrt(math.pi)))
	else:
		tmp = math.fabs(((math.fabs(math.pow(math.fabs(x), 7.0)) * 0.047619047619047616) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 3.1e-7)
		tmp = Float64(abs(abs(x)) * abs(Float64(Float64(2.0 + Float64(0.6666666666666666 * (abs(x) ^ 2.0))) / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(abs((abs(x) ^ 7.0)) * 0.047619047619047616) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 3.1e-7)
		tmp = abs(abs(x)) * abs(((2.0 + (0.6666666666666666 * (abs(x) ^ 2.0))) / sqrt(pi)));
	else
		tmp = abs(((abs((abs(x) ^ 7.0)) * 0.047619047619047616) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 3.1e-7], N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[Abs[N[(N[(2.0 + N[(0.6666666666666666 * N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[Abs[N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-7

   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%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 89.2%

         \[\leadsto \left|x\right| \cdot \left|\frac{2 + 0.6666666666666666 \cdot {x}^{\color{blue}{2}}}{\sqrt{\pi}}\right| \]

   8. Applied rewrites 89.2%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{2 + 0.6666666666666666 \cdot {x}^{2}}}{\sqrt{\pi}}\right| \]

   if 3.1e-7 < 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

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   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 37.5%

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

   6. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      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. pow3 N/A

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

      11. lift-*.f64 N/A

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

      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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 37.5%

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 37.5%

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

Alternative 6: 98.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616 + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $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 99.4%

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 0.047619047619047616, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{3}}, 2\right)\right)}{\sqrt{\pi}}\right| \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

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

Alternative 7: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;\left(2 \cdot \left|\left|x\right|\right|\right) \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left|{\left(\left|x\right|\right)}^{7}\right| \cdot 0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 3.1e-7)
   (* (* 2.0 (fabs (fabs x))) 0.5641895835477563)
   (fabs (/ (* (fabs (pow (fabs x) 7.0)) 0.047619047619047616) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 3.1e-7) {
		tmp = (2.0 * fabs(fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs(((fabs(pow(fabs(x), 7.0)) * 0.047619047619047616) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 3.1e-7) {
		tmp = (2.0 * Math.abs(Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs(((Math.abs(Math.pow(Math.abs(x), 7.0)) * 0.047619047619047616) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 3.1e-7:
		tmp = (2.0 * math.fabs(math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs(((math.fabs(math.pow(math.fabs(x), 7.0)) * 0.047619047619047616) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 3.1e-7)
		tmp = Float64(Float64(2.0 * abs(abs(x))) * 0.5641895835477563);
	else
		tmp = abs(Float64(Float64(abs((abs(x) ^ 7.0)) * 0.047619047619047616) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 3.1e-7)
		tmp = (2.0 * abs(abs(x))) * 0.5641895835477563;
	else
		tmp = abs(((abs((abs(x) ^ 7.0)) * 0.047619047619047616) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 3.1e-7], N[(N[(2.0 * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[Abs[N[(N[(N[Abs[N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-7

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 66.6%

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

   6. Applied rewrites 66.6%

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

   7. Evaluated real constant 66.8%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|2 \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot 0.5641895835477563} \]

   if 3.1e-7 < 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

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   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 37.5%

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

   6. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      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. pow3 N/A

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

      11. lift-*.f64 N/A

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

      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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 37.5%

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 37.5%

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

Alternative 8: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;\left(2 \cdot \left|\left|x\right|\right|\right) \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left|{\left(\left|x\right|\right)}^{7}\right|}{\sqrt{\pi}} \cdot 0.047619047619047616\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 3.1e-7)
   (* (* 2.0 (fabs (fabs x))) 0.5641895835477563)
   (fabs (* (/ (fabs (pow (fabs x) 7.0)) (sqrt PI)) 0.047619047619047616))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 3.1e-7) {
		tmp = (2.0 * fabs(fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs(((fabs(pow(fabs(x), 7.0)) / sqrt(((double) M_PI))) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 3.1e-7) {
		tmp = (2.0 * Math.abs(Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs(((Math.abs(Math.pow(Math.abs(x), 7.0)) / Math.sqrt(Math.PI)) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 3.1e-7:
		tmp = (2.0 * math.fabs(math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs(((math.fabs(math.pow(math.fabs(x), 7.0)) / math.sqrt(math.pi)) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 3.1e-7)
		tmp = Float64(Float64(2.0 * abs(abs(x))) * 0.5641895835477563);
	else
		tmp = abs(Float64(Float64(abs((abs(x) ^ 7.0)) / sqrt(pi)) * 0.047619047619047616));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 3.1e-7)
		tmp = (2.0 * abs(abs(x))) * 0.5641895835477563;
	else
		tmp = abs(((abs((abs(x) ^ 7.0)) / sqrt(pi)) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 3.1e-7], N[(N[(2.0 * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[Abs[N[(N[(N[Abs[N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-7

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 66.6%

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

   6. Applied rewrites 66.6%

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

   7. Evaluated real constant 66.8%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|2 \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot 0.5641895835477563} \]

   if 3.1e-7 < 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

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   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 37.5%

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

   6. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      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. pow3 N/A

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

      11. lift-*.f64 N/A

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

      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

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 37.5%

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

   8. Applied rewrites 37.5%

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

   10. Applied rewrites 37.5%

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

Alternative 9: 83.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\left(2 \cdot \left|\left|x\right|\right|\right) \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2 \cdot \sqrt{\left|x\right| \cdot \left|x\right|}\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e-10)
   (* (* 2.0 (fabs (fabs x))) 0.5641895835477563)
   (/ (fabs (* 2.0 (sqrt (* (fabs x) (fabs x))))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-10) {
		tmp = (2.0 * fabs(fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs((2.0 * sqrt((fabs(x) * fabs(x))))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e-10) {
		tmp = (2.0 * Math.abs(Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((Math.abs(x) * Math.abs(x))))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e-10:
		tmp = (2.0 * math.fabs(math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs((2.0 * math.sqrt((math.fabs(x) * math.fabs(x))))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e-10)
		tmp = Float64(Float64(2.0 * abs(abs(x))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(2.0 * sqrt(Float64(abs(x) * abs(x))))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e-10)
		tmp = (2.0 * abs(abs(x))) * 0.5641895835477563;
	else
		tmp = abs((2.0 * sqrt((abs(x) * abs(x))))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e-10], N[(N[(2.0 * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(2.0 * N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000003e-10

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 66.6%

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

   6. Applied rewrites 66.6%

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

   7. Evaluated real constant 66.8%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|2 \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot 0.5641895835477563} \]

   if 5.0000000000000003e-10 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 66.6%

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

   6. Applied rewrites 66.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 53.4%

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

   8. Applied rewrites 53.4%

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

Alternative 10: 83.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\left(2 \cdot \left|\left|x\right|\right|\right) \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2 \cdot \sqrt{\left|x\right| \cdot \left|x\right|}\right|}{1.772453850905516}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e+148)
   (* (* 2.0 (fabs (fabs x))) 0.5641895835477563)
   (/ (fabs (* 2.0 (sqrt (* (fabs x) (fabs x))))) 1.772453850905516)))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e+148) {
		tmp = (2.0 * fabs(fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs((2.0 * sqrt((fabs(x) * fabs(x))))) / 1.772453850905516;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 5d+148) then
        tmp = (2.0d0 * abs(abs(x))) * 0.5641895835477563d0
    else
        tmp = abs((2.0d0 * sqrt((abs(x) * abs(x))))) / 1.772453850905516d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e+148) {
		tmp = (2.0 * Math.abs(Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((Math.abs(x) * Math.abs(x))))) / 1.772453850905516;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e+148:
		tmp = (2.0 * math.fabs(math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs((2.0 * math.sqrt((math.fabs(x) * math.fabs(x))))) / 1.772453850905516
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e+148)
		tmp = Float64(Float64(2.0 * abs(abs(x))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(2.0 * sqrt(Float64(abs(x) * abs(x))))) / 1.772453850905516);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e+148)
		tmp = (2.0 * abs(abs(x))) * 0.5641895835477563;
	else
		tmp = abs((2.0 * sqrt((abs(x) * abs(x))))) / 1.772453850905516;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e+148], N[(N[(2.0 * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(2.0 * N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e148

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 66.6%

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

   6. Applied rewrites 66.6%

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

   7. Evaluated real constant 66.8%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|2 \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot 0.5641895835477563} \]

   if 5.0000000000000002e148 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 66.6%

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

   6. Applied rewrites 66.6%

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

   7. Evaluated real constant 66.8%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

         \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

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

         \[\leadsto \frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{\frac{7982422502469483}{4503599627370496}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{\frac{7982422502469483}{4503599627370496}} \]

      4. lower-sqrt.f64 53.5%

         \[\leadsto \frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{1.772453850905516} \]

   9. Applied rewrites 53.5%

      \[\leadsto \frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{1.772453850905516} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.0% accurate, 11.7× speedup

Math

\[\left(2 \cdot \left|x\right|\right) \cdot 0.5641895835477563 \]

FPCore

(FPCore (x) :precision binary64 (* (* 2.0 (fabs x)) 0.5641895835477563))

C

double code(double x) {
	return (2.0 * fabs(x)) * 0.5641895835477563;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (2.0d0 * abs(x)) * 0.5641895835477563d0
end function

Java

public static double code(double x) {
	return (2.0 * Math.abs(x)) * 0.5641895835477563;
}

Python

def code(x):
	return (2.0 * math.fabs(x)) * 0.5641895835477563

Julia

function code(x)
	return Float64(Float64(2.0 * abs(x)) * 0.5641895835477563)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * abs(x)) * 0.5641895835477563;
end

Wolfram

code[x_] := N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.5641895835477563), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 66.6%

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

6. Applied rewrites 66.6%

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

7. Evaluated real constant 66.8%

   \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{1.772453850905516}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left|2 \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}}} \]

9. Applied rewrites 67.0%

   \[\leadsto \color{blue}{\left(2 \cdot \left|x\right|\right) \cdot 0.5641895835477563} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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)))))))