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

Percentage Accurate: 99.8% → 99.8%

Time: 5.9s

Alternatives: 9

Speedup: 1.9×

• 71.0% 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, 1.9× speedup

Math

\[\left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   0.5641895835477563
   (fma
    (* (* (fma (* 0.2 x) x 0.6666666666666666) x) x)
    x
    (fma (pow x 7.0) 0.047619047619047616 (+ x x))))))

C

double code(double x) {
	return fabs((0.5641895835477563 * fma(((fma((0.2 * x), x, 0.6666666666666666) * x) * x), x, fma(pow(x, 7.0), 0.047619047619047616, (x + x)))));
}

Julia

function code(x)
	return abs(Float64(0.5641895835477563 * fma(Float64(Float64(fma(Float64(0.2 * x), x, 0.6666666666666666) * x) * x), x, fma((x ^ 7.0), 0.047619047619047616, Float64(x + x)))))
end

Wolfram

code[x_] := N[Abs[N[(0.5641895835477563 * N[(N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-+l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Evaluated real constant 99.8%

   \[\leadsto \left|\color{blue}{0.5641895835477563} \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right)}, \left|x\right|, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot \left(x \cdot x\right)}, \left|x\right|, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot \color{blue}{\left(x \cdot x\right)}, \left|x\right|, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   4. associate-*r* N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x}, \left|x\right|, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x}, \left|x\right|, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   6. lower-*.f64 99.8%

      \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right)} \cdot x, \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   7. lift-fabs.f64 N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, \color{blue}{\left|x\right|}, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

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

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, \color{blue}{\sqrt{x \cdot x}}, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   9. sqrt-unprod N/A

      \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, \mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   10. rem-square-sqrt 73.7%

       \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right) \cdot x, \color{blue}{x}, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   11. lift-fma.f64 N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \color{blue}{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}\right)\right| \]

   12. *-commutative N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \color{blue}{{\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}} + 2 \cdot \left|x\right|\right)\right| \]

   13. lower-fma.f64 73.7%

       \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, 2 \cdot \left|x\right|\right)}\right)\right| \]

   14. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({\color{blue}{\left(\left|x\right|\right)}}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right)\right| \]

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

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right)\right| \]

   16. sqrt-unprod N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right)\right| \]

   17. rem-square-sqrt 99.3%

       \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({\color{blue}{x}}^{7}, 0.047619047619047616, 2 \cdot \left|x\right|\right)\right)\right| \]

   18. lift-*.f64 N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \color{blue}{2 \cdot \left|x\right|}\right)\right)\right| \]

   19. lift-fabs.f64 N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \color{blue}{\left|x\right|}\right)\right)\right| \]

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

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \color{blue}{\sqrt{x \cdot x}}\right)\right)\right| \]

   21. sqrt-unprod N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)\right| \]

   22. rem-square-sqrt N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \color{blue}{x}\right)\right)\right| \]

   23. count-2-rev N/A

       \[\leadsto \left|\frac{5081767996463981}{9007199254740992} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \color{blue}{x + x}\right)\right)\right| \]

   24. lower-+.f64 99.8%

       \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, 0.047619047619047616, \color{blue}{x + x}\right)\right)\right| \]

8. Applied rewrites 99.8%

   \[\leadsto \left|0.5641895835477563 \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot x\right) \cdot x, x, \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right)}\right| \]
9. Add Preprocessing

Alternative 2: 99.1% accurate, 2.1× speedup

Math

\[\left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.6666666666666666, \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   0.5641895835477563
   (fma
    (* (* x x) 0.6666666666666666)
    (fabs x)
    (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x)))))))

C

double code(double x) {
	return fabs((0.5641895835477563 * fma(((x * x) * 0.6666666666666666), fabs(x), fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x))))));
}

Julia

function code(x)
	return abs(Float64(0.5641895835477563 * fma(Float64(Float64(x * x) * 0.6666666666666666), abs(x), fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x))))))
end

Wolfram

code[x_] := N[Abs[N[(0.5641895835477563 * N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-+l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Evaluated real constant 99.8%

   \[\leadsto \left|\color{blue}{0.5641895835477563} \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

7. Taylor expanded in x around 0

   \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{2}{3}}, \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]
8. Step-by-step derivation

9. Applied rewrites 99.1%

   \[\leadsto \left|0.5641895835477563 \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{0.6666666666666666}, \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]
10. Add Preprocessing

Alternative 3: 98.8% accurate, 2.9× speedup

Math

\[\left|0.5641895835477563 \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   0.5641895835477563
   (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x))))))

C

double code(double x) {
	return fabs((0.5641895835477563 * fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x)))));
}

Julia

function code(x)
	return abs(Float64(0.5641895835477563 * fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x)))))
end

Wolfram

code[x_] := N[Abs[N[(0.5641895835477563 * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 98.8%

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

6. Applied rewrites 98.8%

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

7. Evaluated real constant 98.8%

   \[\leadsto \left|\color{blue}{0.5641895835477563} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right| \]
8. Add Preprocessing

Alternative 4: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 11500000000:\\ \;\;\;\;\left|\left|x\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|{\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 11500000000.0)
   (fabs (* (fabs x) 1.1283791670955126))
   (* (/ 1.0 (sqrt PI)) (fabs (* (pow (fabs x) 7.0) 0.047619047619047616)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 11500000000.0) {
		tmp = fabs((fabs(x) * 1.1283791670955126));
	} else {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((pow(fabs(x), 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 11500000000.0) {
		tmp = Math.abs((Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((Math.pow(Math.abs(x), 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 11500000000.0:
		tmp = math.fabs((math.fabs(x) * 1.1283791670955126))
	else:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((math.pow(math.fabs(x), 7.0) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 11500000000.0)
		tmp = abs(Float64(abs(x) * 1.1283791670955126));
	else
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64((abs(x) ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 11500000000.0)
		tmp = abs((abs(x) * 1.1283791670955126));
	else
		tmp = (1.0 / sqrt(pi)) * abs(((abs(x) ^ 7.0) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 11500000000.0], N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e10

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

   7. Evaluated real constant 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-*r/ N/A

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

      8. count-2-rev N/A

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

      9. div-add-rev N/A

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

      10. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{x}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      11. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + x \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      12. distribute-lft-out N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      14. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      16. metadata-eval 67.2%

          \[\leadsto \left|x \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.2%

      \[\leadsto \left|x \cdot \color{blue}{1.1283791670955126}\right| \]

   if 1.15e10 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 37.3%

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

   6. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow3 N/A

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

      8. pow3 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 37.3%

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

   8. Applied rewrites 37.3%

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

   10. Applied rewrites 37.3%

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

Alternative 5: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 11500000000:\\ \;\;\;\;\left|\left|x\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 11500000000.0)
   (fabs (* (fabs x) 1.1283791670955126))
   (/ (fabs (* (pow (fabs x) 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 11500000000.0) {
		tmp = fabs((fabs(x) * 1.1283791670955126));
	} else {
		tmp = 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) <= 11500000000.0) {
		tmp = Math.abs((Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = 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) <= 11500000000.0:
		tmp = math.fabs((math.fabs(x) * 1.1283791670955126))
	else:
		tmp = 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) <= 11500000000.0)
		tmp = abs(Float64(abs(x) * 1.1283791670955126));
	else
		tmp = Float64(abs(Float64((abs(x) ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 11500000000.0)
		tmp = abs((abs(x) * 1.1283791670955126));
	else
		tmp = abs(((abs(x) ^ 7.0) * 0.047619047619047616)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 11500000000.0], N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e10

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

   7. Evaluated real constant 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-*r/ N/A

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

      8. count-2-rev N/A

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

      9. div-add-rev N/A

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

      10. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{x}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      11. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + x \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      12. distribute-lft-out N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      14. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      16. metadata-eval 67.2%

          \[\leadsto \left|x \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.2%

      \[\leadsto \left|x \cdot \color{blue}{1.1283791670955126}\right| \]

   if 1.15e10 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 37.3%

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

   6. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow3 N/A

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

      8. pow3 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 37.3%

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

   8. Applied rewrites 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

   10. Applied rewrites 37.3%

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

Alternative 6: 92.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 11500000000:\\ \;\;\;\;\left|\left|x\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.11283791670955126 \cdot \left({\left(\left|x\right|\right)}^{4} \cdot \left|\left|x\right|\right|\right)\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 11500000000.0)
   (fabs (* (fabs x) 1.1283791670955126))
   (fabs (* 0.11283791670955126 (* (pow (fabs x) 4.0) (fabs (fabs x)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 11500000000.0) {
		tmp = fabs((fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((0.11283791670955126 * (pow(fabs(x), 4.0) * fabs(fabs(x)))));
	}
	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) <= 11500000000.0d0) then
        tmp = abs((abs(x) * 1.1283791670955126d0))
    else
        tmp = abs((0.11283791670955126d0 * ((abs(x) ** 4.0d0) * abs(abs(x)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 11500000000.0) {
		tmp = Math.abs((Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((0.11283791670955126 * (Math.pow(Math.abs(x), 4.0) * Math.abs(Math.abs(x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 11500000000.0:
		tmp = math.fabs((math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((0.11283791670955126 * (math.pow(math.fabs(x), 4.0) * math.fabs(math.fabs(x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 11500000000.0)
		tmp = abs(Float64(abs(x) * 1.1283791670955126));
	else
		tmp = abs(Float64(0.11283791670955126 * Float64((abs(x) ^ 4.0) * abs(abs(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 11500000000.0)
		tmp = abs((abs(x) * 1.1283791670955126));
	else
		tmp = abs((0.11283791670955126 * ((abs(x) ^ 4.0) * abs(abs(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 11500000000.0], N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.11283791670955126 * N[(N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision] * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e10

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

   7. Evaluated real constant 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-*r/ N/A

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

      8. count-2-rev N/A

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

      9. div-add-rev N/A

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

      10. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{x}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      11. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + x \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      12. distribute-lft-out N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      14. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      16. metadata-eval 67.2%

          \[\leadsto \left|x \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.2%

      \[\leadsto \left|x \cdot \color{blue}{1.1283791670955126}\right| \]

   if 1.15e10 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Evaluated real constant 99.8%

      \[\leadsto \left|\color{blue}{0.5641895835477563} \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), \left|x\right|, \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)\right)\right| \]

   7. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{5081767996463981}{45035996273704960} \cdot \left({x}^{4} \cdot \left|x\right|\right)}\right| \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{5081767996463981}{45035996273704960} \cdot \color{blue}{\left({x}^{4} \cdot \left|x\right|\right)}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{5081767996463981}{45035996273704960} \cdot \left({x}^{4} \cdot \color{blue}{\left|x\right|}\right)\right| \]

      3. lower-pow.f64 N/A

         \[\leadsto \left|\frac{5081767996463981}{45035996273704960} \cdot \left({x}^{4} \cdot \left|\color{blue}{x}\right|\right)\right| \]

      4. lower-fabs.f64 31.5%

         \[\leadsto \left|0.11283791670955126 \cdot \left({x}^{4} \cdot \left|x\right|\right)\right| \]

   9. Applied rewrites 31.5%

      \[\leadsto \left|\color{blue}{0.11283791670955126 \cdot \left({x}^{4} \cdot \left|x\right|\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left|\left|x\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{\left|x\right| \cdot \left|x\right|}{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-7)
   (fabs (* (fabs x) 1.1283791670955126))
   (fabs (* 2.0 (sqrt (/ (* (fabs x) (fabs x)) PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-7) {
		tmp = fabs((fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * sqrt(((fabs(x) * fabs(x)) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-7) {
		tmp = Math.abs((Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((Math.abs(x) * Math.abs(x)) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-7:
		tmp = math.fabs((math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((math.fabs(x) * math.fabs(x)) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-7)
		tmp = abs(Float64(abs(x) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(abs(x) * abs(x)) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-7)
		tmp = abs((abs(x) * 1.1283791670955126));
	else
		tmp = abs((2.0 * sqrt(((abs(x) * abs(x)) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-7], N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-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 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

   7. Evaluated real constant 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-*r/ N/A

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

      8. count-2-rev N/A

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

      9. div-add-rev N/A

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

      10. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{x}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      11. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + x \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      12. distribute-lft-out N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      14. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      16. metadata-eval 67.2%

          \[\leadsto \left|x \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.2%

      \[\leadsto \left|x \cdot \color{blue}{1.1283791670955126}\right| \]

   if 1.9999999999999999e-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 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 53.7%

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

   8. Applied rewrites 53.7%

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

Alternative 8: 83.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot 1.1283791670955126\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0 \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) 1.1283791670955126)))
   (if (<= (fabs x) 5e+146) (fabs t_0) (sqrt (* t_0 t_0)))))

C

double code(double x) {
	double t_0 = fabs(x) * 1.1283791670955126;
	double tmp;
	if (fabs(x) <= 5e+146) {
		tmp = fabs(t_0);
	} else {
		tmp = sqrt((t_0 * t_0));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = abs(x) * 1.1283791670955126d0
    if (abs(x) <= 5d+146) then
        tmp = abs(t_0)
    else
        tmp = sqrt((t_0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * 1.1283791670955126;
	double tmp;
	if (Math.abs(x) <= 5e+146) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.sqrt((t_0 * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * 1.1283791670955126
	tmp = 0
	if math.fabs(x) <= 5e+146:
		tmp = math.fabs(t_0)
	else:
		tmp = math.sqrt((t_0 * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * 1.1283791670955126)
	tmp = 0.0
	if (abs(x) <= 5e+146)
		tmp = abs(t_0);
	else
		tmp = sqrt(Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * 1.1283791670955126;
	tmp = 0.0;
	if (abs(x) <= 5e+146)
		tmp = abs(t_0);
	else
		tmp = sqrt((t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e+146], N[Abs[t$95$0], $MachinePrecision], N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e146

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

   7. Evaluated real constant 67.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fabs.f64 N/A

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

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

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

      5. sqrt-unprod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-*r/ N/A

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

      8. count-2-rev N/A

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

      9. div-add-rev N/A

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

      10. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{x}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      11. mult-flip N/A

          \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + x \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      12. distribute-lft-out N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      14. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      15. metadata-eval N/A

          \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      16. metadata-eval 67.2%

          \[\leadsto \left|x \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.2%

      \[\leadsto \left|x \cdot \color{blue}{1.1283791670955126}\right| \]

   if 4.9999999999999999e146 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.8%

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

   6. Applied rewrites 66.8%

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

   7. Evaluated real constant 67.0%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

   9. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 1.1283791670955126\right) \cdot \left(x \cdot 1.1283791670955126\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.2% accurate, 18.8× speedup

Math

\[\left|x \cdot 1.1283791670955126\right| \]

FPCore

(FPCore (x) :precision binary64 (fabs (* x 1.1283791670955126)))

C

double code(double x) {
	return fabs((x * 1.1283791670955126));
}

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 = abs((x * 1.1283791670955126d0))
end function

Java

public static double code(double x) {
	return Math.abs((x * 1.1283791670955126));
}

Python

def code(x):
	return math.fabs((x * 1.1283791670955126))

Julia

function code(x)
	return abs(Float64(x * 1.1283791670955126))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * 1.1283791670955126));
end

Wolfram

code[x_] := N[Abs[N[(x * 1.1283791670955126), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 66.8%

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

6. Applied rewrites 66.8%

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

7. Evaluated real constant 67.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fabs.f64 N/A

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

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

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

   5. sqrt-unprod N/A

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

   6. rem-square-sqrt N/A

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

   7. associate-*r/ N/A

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

   8. count-2-rev N/A

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

   9. div-add-rev N/A

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

   10. mult-flip N/A

       \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{x}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   11. mult-flip N/A

       \[\leadsto \left|x \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + x \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   12. distribute-lft-out N/A

       \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   13. lower-*.f64 N/A

       \[\leadsto \left|x \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   14. metadata-eval N/A

       \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   15. metadata-eval N/A

       \[\leadsto \left|x \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

   16. metadata-eval 67.2%

       \[\leadsto \left|x \cdot 1.1283791670955126\right| \]

9. Applied rewrites 67.2%

   \[\leadsto \left|x \cdot \color{blue}{1.1283791670955126}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025209 
(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)))))))