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

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 13

Speedup: N/A×

• 72.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8%

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. associate-+r+ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-fma.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8%

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. associate-+r+ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-fma.f64 N/A

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-fabs.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. mul-fabs N/A

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

   5. lower-fabs.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8%

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. associate-+r+ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-fma.f64 N/A

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.3%

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

Alternative 4: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 17:\\ \;\;\;\;\left|\left(-\left|x\right|\right) - \left|x\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{6} \cdot \left|\left|x\right|\right|\right)\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 17.0)
   (* (fabs (- (- (fabs x)) (fabs x))) 0.5641895835477563)
   (/
    (fabs (* 0.047619047619047616 (* (pow (fabs x) 6.0) (fabs (fabs x)))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 17.0) {
		tmp = fabs((-fabs(x) - fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs((0.047619047619047616 * (pow(fabs(x), 6.0) * fabs(fabs(x))))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 17.0) {
		tmp = Math.abs((-Math.abs(x) - Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.abs(x), 6.0) * Math.abs(Math.abs(x))))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 17.0:
		tmp = math.fabs((-math.fabs(x) - math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.fabs(x), 6.0) * math.fabs(math.fabs(x))))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 17.0)
		tmp = Float64(abs(Float64(Float64(-abs(x)) - abs(x))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * Float64((abs(x) ^ 6.0) * abs(abs(x))))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 17.0)
		tmp = abs((-abs(x) - abs(x))) * 0.5641895835477563;
	else
		tmp = abs((0.047619047619047616 * ((abs(x) ^ 6.0) * abs(abs(x))))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 17.0], N[(N[Abs[N[((-N[Abs[x], $MachinePrecision]) - N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 17 < 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 35.3%

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

   6. Applied rewrites 35.3%

      \[\leadsto \frac{\left|\color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right|}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 17:\\ \;\;\;\;\left|\left(-\left|x\right|\right) - \left|x\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{\left(\left|x\right|\right)}^{6} \cdot \left|\left|x\right|\right|}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 17.0)
   (* (fabs (- (- (fabs x)) (fabs x))) 0.5641895835477563)
   (fabs
    (*
     0.047619047619047616
     (/ (* (pow (fabs x) 6.0) (fabs (fabs x))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 17.0) {
		tmp = fabs((-fabs(x) - fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(fabs(x), 6.0) * fabs(fabs(x))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 17.0) {
		tmp = Math.abs((-Math.abs(x) - Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((0.047619047619047616 * ((Math.pow(Math.abs(x), 6.0) * Math.abs(Math.abs(x))) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 17.0:
		tmp = math.fabs((-math.fabs(x) - math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs((0.047619047619047616 * ((math.pow(math.fabs(x), 6.0) * math.fabs(math.fabs(x))) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 17.0)
		tmp = Float64(abs(Float64(Float64(-abs(x)) - abs(x))) * 0.5641895835477563);
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((abs(x) ^ 6.0) * abs(abs(x))) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 17.0)
		tmp = abs((-abs(x) - abs(x))) * 0.5641895835477563;
	else
		tmp = abs((0.047619047619047616 * (((abs(x) ^ 6.0) * abs(abs(x))) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 17.0], N[(N[Abs[N[((-N[Abs[x], $MachinePrecision]) - N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 17 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 35.3%

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

   6. Applied rewrites 35.3%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 17:\\ \;\;\;\;\left|\left(-\left|x\right|\right) - \left|x\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left|\left|x\right|\right| \cdot \left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.047619047619047616\right)\right) \cdot \left|x\right|\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 17.0)
   (* (fabs (- (- (fabs x)) (fabs x))) 0.5641895835477563)
   (/
    (fabs
     (*
      (*
       (fabs (fabs x))
       (*
        (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))
        0.047619047619047616))
      (fabs x)))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 17.0) {
		tmp = fabs((-fabs(x) - fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs(((fabs(fabs(x)) * (((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * 0.047619047619047616)) * fabs(x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 17.0) {
		tmp = Math.abs((-Math.abs(x) - Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs(((Math.abs(Math.abs(x)) * (((((Math.abs(x) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)) * 0.047619047619047616)) * Math.abs(x))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 17.0:
		tmp = math.fabs((-math.fabs(x) - math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs(((math.fabs(math.fabs(x)) * (((((math.fabs(x) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)) * 0.047619047619047616)) * math.fabs(x))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 17.0)
		tmp = Float64(abs(Float64(Float64(-abs(x)) - abs(x))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(Float64(abs(abs(x)) * Float64(Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * 0.047619047619047616)) * abs(x))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 17.0)
		tmp = abs((-abs(x) - abs(x))) * 0.5641895835477563;
	else
		tmp = abs(((abs(abs(x)) * (((((abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * 0.047619047619047616)) * abs(x))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 17.0], N[(N[Abs[N[((-N[Abs[x], $MachinePrecision]) - N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 17 < 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 35.3%

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

   6. Applied rewrites 35.3%

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

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. pow-prod-up N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

   8. Applied rewrites 35.3%

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

Alternative 7: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := -\left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 17:\\ \;\;\;\;\left|t\_1 - \left|x\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(0.047619047619047616 \cdot t\_0\right) \cdot \left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) \cdot t\_1\right|}{1.772453850905516}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (fabs x))) (t_1 (- (fabs x))))
   (if (<= (fabs x) 17.0)
     (* (fabs (- t_1 (fabs x))) 0.5641895835477563)
     (/
      (fabs
       (* (* (* 0.047619047619047616 t_0) (* (* t_0 (fabs x)) (fabs x))) t_1))
      1.772453850905516))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = -fabs(x);
	double tmp;
	if (fabs(x) <= 17.0) {
		tmp = fabs((t_1 - fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs((((0.047619047619047616 * t_0) * ((t_0 * fabs(x)) * fabs(x))) * t_1)) / 1.772453850905516;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(x) * abs(x)
    t_1 = -abs(x)
    if (abs(x) <= 17.0d0) then
        tmp = abs((t_1 - abs(x))) * 0.5641895835477563d0
    else
        tmp = abs((((0.047619047619047616d0 * t_0) * ((t_0 * abs(x)) * abs(x))) * t_1)) / 1.772453850905516d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = -Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 17.0) {
		tmp = Math.abs((t_1 - Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((((0.047619047619047616 * t_0) * ((t_0 * Math.abs(x)) * Math.abs(x))) * t_1)) / 1.772453850905516;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = -math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 17.0:
		tmp = math.fabs((t_1 - math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs((((0.047619047619047616 * t_0) * ((t_0 * math.fabs(x)) * math.fabs(x))) * t_1)) / 1.772453850905516
	return tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(-abs(x))
	tmp = 0.0
	if (abs(x) <= 17.0)
		tmp = Float64(abs(Float64(t_1 - abs(x))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(Float64(Float64(0.047619047619047616 * t_0) * Float64(Float64(t_0 * abs(x)) * abs(x))) * t_1)) / 1.772453850905516);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	t_1 = -abs(x);
	tmp = 0.0;
	if (abs(x) <= 17.0)
		tmp = abs((t_1 - abs(x))) * 0.5641895835477563;
	else
		tmp = abs((((0.047619047619047616 * t_0) * ((t_0 * abs(x)) * abs(x))) * t_1)) / 1.772453850905516;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[x], $MachinePrecision])}, If[LessEqual[N[Abs[x], $MachinePrecision], 17.0], N[(N[Abs[N[(t$95$1 - N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(N[(N[(0.047619047619047616 * t$95$0), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 17 < 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 35.3%

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

   6. Applied rewrites 35.3%

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

   7. Evaluated real constant 35.3%

      \[\leadsto \frac{\left|0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|\frac{1}{21} \cdot \color{blue}{\left({x}^{6} \cdot \left|x\right|\right)}\right|}{\frac{7982422502469483}{4503599627370496}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|\frac{1}{21} \cdot \left({x}^{6} \cdot \color{blue}{\left|x\right|}\right)\right|}{\frac{7982422502469483}{4503599627370496}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \color{blue}{\left|x\right|}\right|}{\frac{7982422502469483}{4503599627370496}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot {x}^{\left(3 + 3\right)}\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      6. pow-prod-up N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      7. pow3 N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      8. pow3 N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      15. associate-*l* N/A

          \[\leadsto \frac{\left|\left(\left(\frac{1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot \left|\color{blue}{x}\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      16. *-commutative N/A

          \[\leadsto \frac{\left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21}\right) \cdot x\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21}\right) \cdot x\right) \cdot \left|x\right|\right|}{\frac{7982422502469483}{4503599627370496}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21}\right) \cdot x\right) \cdot \color{blue}{\left|x\right|}\right|}{\frac{7982422502469483}{4503599627370496}} \]

   9. Applied rewrites 35.3%

      \[\leadsto \frac{\left|\left(\left(0.047619047619047616 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(-x\right)}\right|}{1.772453850905516} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := -\left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 17:\\ \;\;\;\;\left|t\_0 - \left|x\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\left|{t\_0}^{7} \cdot 0.047619047619047616\right| \cdot 0.5641895835477563\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (fabs x))))
   (if (<= (fabs x) 17.0)
     (* (fabs (- t_0 (fabs x))) 0.5641895835477563)
     (* (fabs (* (pow t_0 7.0) 0.047619047619047616)) 0.5641895835477563))))

C

double code(double x) {
	double t_0 = -fabs(x);
	double tmp;
	if (fabs(x) <= 17.0) {
		tmp = fabs((t_0 - fabs(x))) * 0.5641895835477563;
	} else {
		tmp = fabs((pow(t_0, 7.0) * 0.047619047619047616)) * 0.5641895835477563;
	}
	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)
    if (abs(x) <= 17.0d0) then
        tmp = abs((t_0 - abs(x))) * 0.5641895835477563d0
    else
        tmp = abs(((t_0 ** 7.0d0) * 0.047619047619047616d0)) * 0.5641895835477563d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 17.0) {
		tmp = Math.abs((t_0 - Math.abs(x))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((Math.pow(t_0, 7.0) * 0.047619047619047616)) * 0.5641895835477563;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 17.0:
		tmp = math.fabs((t_0 - math.fabs(x))) * 0.5641895835477563
	else:
		tmp = math.fabs((math.pow(t_0, 7.0) * 0.047619047619047616)) * 0.5641895835477563
	return tmp

Julia

function code(x)
	t_0 = Float64(-abs(x))
	tmp = 0.0
	if (abs(x) <= 17.0)
		tmp = Float64(abs(Float64(t_0 - abs(x))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64((t_0 ^ 7.0) * 0.047619047619047616)) * 0.5641895835477563);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -abs(x);
	tmp = 0.0;
	if (abs(x) <= 17.0)
		tmp = abs((t_0 - abs(x))) * 0.5641895835477563;
	else
		tmp = abs(((t_0 ^ 7.0) * 0.047619047619047616)) * 0.5641895835477563;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = (-N[Abs[x], $MachinePrecision])}, If[LessEqual[N[Abs[x], $MachinePrecision], 17.0], N[(N[Abs[N[(t$95$0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(N[Power[t$95$0, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 17

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 17 < 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 35.3%

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

   6. Applied rewrites 35.3%

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

   7. Evaluated real constant 35.3%

      \[\leadsto \frac{\left|0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|}{\color{blue}{1.772453850905516}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left|\frac{1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|}{\frac{7982422502469483}{4503599627370496}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|\frac{1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|\frac{1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\left|{\left(-x\right)}^{7} \cdot 0.047619047619047616\right| \cdot 0.5641895835477563} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.7% accurate, 3.0× speedup

Math

\[\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x)))
   1.772453850905516)))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x))) / 1.772453850905516));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x))) / 1.772453850905516))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-PI.f64 98.4%

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

6. Applied rewrites 98.4%

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

7. Evaluated real constant 98.7%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]
8. Add Preprocessing

Alternative 10: 89.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 89.5%

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

6. Applied rewrites 89.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-rgt-out N/A

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 89.5%

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

8. Applied rewrites 89.5%

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

Alternative 11: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 10^{-8}:\\ \;\;\;\;\left|\left(-x\right) - x\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          (/ 1.0 (sqrt PI))
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
           (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))
        1e-8)
     (* (fabs (- (- x) x)) 0.5641895835477563)
     (/ (fabs (* 2.0 (sqrt (* x x)))) (sqrt PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x)))))) <= 1e-8) {
		tmp = fabs((-x - x)) * 0.5641895835477563;
	} else {
		tmp = fabs((2.0 * sqrt((x * x)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x)))))) <= 1e-8) {
		tmp = Math.abs((-x - x)) * 0.5641895835477563;
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((x * x)))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x)))))) <= 1e-8:
		tmp = math.fabs((-x - x)) * 0.5641895835477563
	else:
		tmp = math.fabs((2.0 * math.sqrt((x * x)))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x)))))) <= 1e-8)
		tmp = Float64(abs(Float64(Float64(-x) - x)) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(2.0 * sqrt(Float64(x * x)))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x)))))) <= 1e-8)
		tmp = abs((-x - x)) * 0.5641895835477563;
	else
		tmp = abs((2.0 * sqrt((x * x)))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-8], N[(N[Abs[N[((-x) - x), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(2.0 * N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 1e-8

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 1e-8 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 53.2%

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

   8. Applied rewrites 53.2%

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

Alternative 12: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 10^{+19}:\\ \;\;\;\;\left|\left(-x\right) - x\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{1.772453850905516}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          (/ 1.0 (sqrt PI))
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
           (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))
        1e+19)
     (* (fabs (- (- x) x)) 0.5641895835477563)
     (/ (fabs (* 2.0 (sqrt (* x x)))) 1.772453850905516))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x)))))) <= 1e+19) {
		tmp = fabs((-x - x)) * 0.5641895835477563;
	} else {
		tmp = fabs((2.0 * sqrt((x * x)))) / 1.772453850905516;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x)))))) <= 1e+19) {
		tmp = Math.abs((-x - x)) * 0.5641895835477563;
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((x * x)))) / 1.772453850905516;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x)))))) <= 1e+19:
		tmp = math.fabs((-x - x)) * 0.5641895835477563
	else:
		tmp = math.fabs((2.0 * math.sqrt((x * x)))) / 1.772453850905516
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x)))))) <= 1e+19)
		tmp = Float64(abs(Float64(Float64(-x) - x)) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(2.0 * sqrt(Float64(x * x)))) / 1.772453850905516);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x)))))) <= 1e+19)
		tmp = abs((-x - x)) * 0.5641895835477563;
	else
		tmp = abs((2.0 * sqrt((x * x)))) / 1.772453850905516;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+19], N[(N[Abs[N[((-x) - x), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(2.0 * N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 1e19

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   9. Applied rewrites 69.4%

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

   if 1e19 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 68.9%

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

   6. Applied rewrites 68.9%

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

   7. Evaluated real constant 69.2%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 53.3%

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

   9. Applied rewrites 53.3%

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

Alternative 13: 69.4% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 68.9%

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

6. Applied rewrites 68.9%

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

7. Evaluated real constant 69.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

9. Applied rewrites 69.4%

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

Reproduce

herbie shell --seed 2025193 
(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)))))))