Rust f64::asinh

Percentage Accurate: 30.3% → 98.9%

Time: 6.7s

Alternatives: 12

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Initial Program: 30.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

double code(double x) {
	return copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
}

Python

def code(x):
	return math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)

Julia

function code(x)
	return copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(0.075 \cdot {x}^{5} + \left(-0.16666666666666666 \cdot {x}^{3} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -20.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.005)
       (copysign
        (+
         (* 0.075 (pow x 5.0))
         (+
          (* -0.16666666666666666 (pow x 3.0))
          (+ x (* -0.044642857142857144 (pow x 7.0)))))
        x)
       (copysign (log (+ x x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(((0.075 * pow(x, 5.0)) + ((-0.16666666666666666 * pow(x, 3.0)) + (x + (-0.044642857142857144 * pow(x, 7.0))))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 0.005) {
		tmp = Math.copySign(((0.075 * Math.pow(x, 5.0)) + ((-0.16666666666666666 * Math.pow(x, 3.0)) + (x + (-0.044642857142857144 * Math.pow(x, 7.0))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -20.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 0.005:
		tmp = math.copysign(((0.075 * math.pow(x, 5.0)) + ((-0.16666666666666666 * math.pow(x, 3.0)) + (x + (-0.044642857142857144 * math.pow(x, 7.0))))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(Float64(Float64(0.075 * (x ^ 5.0)) + Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(x + Float64(-0.044642857142857144 * (x ^ 7.0))))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (t_0 <= 0.005)
		tmp = sign(x) * abs(((0.075 * (x ^ 5.0)) + ((-0.16666666666666666 * (x ^ 3.0)) + (x + (-0.044642857142857144 * (x ^ 7.0))))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[With[{TMP1 = Abs[N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -20

   1. Initial program 51.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + -1 \cdot x\right)}, x\right) \]
   3. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| - x\right)}, x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| - x\right)}, x\right) \]

   if -20 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 0.0050000000000000001

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 9.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 9.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 9.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 9.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 9.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x + \left(-0.5 \cdot {x}^{3} + \left(-0.13392857142857142 \cdot {x}^{7} + 0.225 \cdot {x}^{5}\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.075 \cdot {x}^{5} + \left(-0.16666666666666666 \cdot {x}^{3} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}, x\right) \]

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(0.075 \cdot {x}^{5} + \left(-0.16666666666666666 \cdot {x}^{3} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.01)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.005)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (log (+ x x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(((-0.16666666666666666 * pow(x, 3.0)) + (x + (0.075 * pow(x, 5.0)))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.005) {
		tmp = Math.copySign(((-0.16666666666666666 * Math.pow(x, 3.0)) + (x + (0.075 * Math.pow(x, 5.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -0.01:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 0.005:
		tmp = math.copysign(((-0.16666666666666666 * math.pow(x, 3.0)) + (x + (0.075 * math.pow(x, 5.0)))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(x + Float64(0.075 * (x ^ 5.0)))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 0.005)
		tmp = sign(x) * abs(((-0.16666666666666666 * (x ^ 3.0)) + (x + (0.075 * (x ^ 5.0)))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[With[{TMP1 = Abs[N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -0.0100000000000000002

   1. Initial program 52.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 28.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 28.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 28.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 28.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 52.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 4.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. frac-2neg 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left(x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      3. log-div 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log \left(-\left(x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]

      4. hypot-udef 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      5. hypot-udef 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      6. add-sqr-sqrt 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\left(1 \cdot 1 + x \cdot x\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \left(\color{blue}{1} + x \cdot x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      8. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      9. fma-def 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

   5. Applied egg-rr 3.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. neg-sub0 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(0 - \left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      2. associate--r- 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\left(0 - x \cdot x\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      3. neg-sub0 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(-x \cdot x\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      4. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-x \cdot x\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      5. sub-neg 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      6. sub-neg 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-x \cdot x\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      7. fma-udef 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(x \cdot x + 1\right)} + \left(-x \cdot x\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      8. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(1 + x \cdot x\right)} + \left(-x \cdot x\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      9. associate-+l+ 50.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(1 + \left(x \cdot x + \left(-x \cdot x\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      10. sub-neg 50.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(1 + \color{blue}{\left(x \cdot x - x \cdot x\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      11. +-inverses 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(1 + \color{blue}{0}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{1} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      14. neg-sub0 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]

      15. neg-sub0 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}\right)\right), x\right) \]

      16. associate--r- 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \left(\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      18. +-commutative 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) + \left(-x\right)\right)}\right)\right), x\right) \]

      19. sub-neg 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) - x\right)}\right)\right), x\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}\right), x\right) \]
   8. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}, x\right) \]

      2. metadata-eval 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1} \cdot \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right), x\right) \]

      3. *-un-lft-identity 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

      4. neg-sub0 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]
   10. Step-by-step derivation

       1. neg-sub0 99.9%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   11. Simplified 99.9%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.0100000000000000002 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 0.0050000000000000001

   1. Initial program 8.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 8.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 8.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 8.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 8.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 8.6%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 8.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. frac-2neg 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left(x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      3. log-div 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log \left(-\left(x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]

      4. hypot-udef 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      5. hypot-udef 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      6. add-sqr-sqrt 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\left(1 \cdot 1 + x \cdot x\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \left(\color{blue}{1} + x \cdot x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      8. +-commutative 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      9. fma-def 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

   5. Applied egg-rr 8.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. neg-sub0 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(0 - \left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      2. associate--r- 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\left(0 - x \cdot x\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      3. neg-sub0 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(-x \cdot x\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      4. +-commutative 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-x \cdot x\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      5. sub-neg 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      6. sub-neg 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-x \cdot x\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      7. fma-udef 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(x \cdot x + 1\right)} + \left(-x \cdot x\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      8. +-commutative 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(1 + x \cdot x\right)} + \left(-x \cdot x\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      9. associate-+l+ 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(1 + \left(x \cdot x + \left(-x \cdot x\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      10. sub-neg 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(1 + \color{blue}{\left(x \cdot x - x \cdot x\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      11. +-inverses 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(1 + \color{blue}{0}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{1} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      14. neg-sub0 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]

      15. neg-sub0 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}\right)\right), x\right) \]

      16. associate--r- 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \left(\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      18. +-commutative 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) + \left(-x\right)\right)}\right)\right), x\right) \]

      19. sub-neg 8.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) - x\right)}\right)\right), x\right) \]

   7. Simplified 8.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}\right), x\right) \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.01:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 0.001)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (x <= 0.001) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (x <= 0.001) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif x <= 0.001:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (x <= 0.001)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= 0.001)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.001], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 51.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + -1 \cdot x\right)}, x\right) \]
   3. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| - x\right)}, x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| - x\right)}, x\right) \]

   if -1.26000000000000001 < x < 1e-3

   1. Initial program 8.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 8.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 8.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 8.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 8.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 3.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 3.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 8.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 8.7%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 8.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right) \]

   if 1e-3 < x

   1. Initial program 55.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 55.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity 99.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00105)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.001)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (x <= 0.001) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 0.001) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.00105:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 0.001:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.00105)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (x <= 0.001)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00105)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (x <= 0.001)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.00105], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.001], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00104999999999999994

   1. Initial program 52.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 28.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 28.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 28.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 28.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 52.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 4.5%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. flip-+ 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}\right), x\right) \]

      2. frac-2neg 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-\left(x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}\right), x\right) \]

      3. log-div 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log \left(-\left(x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]

      4. hypot-udef 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      5. hypot-udef 2.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      6. add-sqr-sqrt 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\left(1 \cdot 1 + x \cdot x\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      7. metadata-eval 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \left(\color{blue}{1} + x \cdot x\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      8. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      9. fma-def 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(-\left(x \cdot x - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

   5. Applied egg-rr 3.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. neg-sub0 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(0 - \left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      2. associate--r- 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\left(0 - x \cdot x\right) + \mathsf{fma}\left(x, x, 1\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      3. neg-sub0 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(-x \cdot x\right)} + \mathsf{fma}\left(x, x, 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      4. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-x \cdot x\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      5. sub-neg 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - x \cdot x\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      6. sub-neg 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) + \left(-x \cdot x\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      7. fma-udef 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(x \cdot x + 1\right)} + \left(-x \cdot x\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      8. +-commutative 3.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(\color{blue}{\left(1 + x \cdot x\right)} + \left(-x \cdot x\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      9. associate-+l+ 50.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{\left(1 + \left(x \cdot x + \left(-x \cdot x\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      10. sub-neg 50.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(1 + \color{blue}{\left(x \cdot x - x \cdot x\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      11. +-inverses 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \left(1 + \color{blue}{0}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      12. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\log \color{blue}{1} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(\color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right), x\right) \]

      14. neg-sub0 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)\right)}\right), x\right) \]

      15. neg-sub0 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}\right)\right), x\right) \]

      16. associate--r- 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)\right)}\right)\right), x\right) \]

      17. neg-sub0 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \left(\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      18. +-commutative 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) + \left(-x\right)\right)}\right)\right), x\right) \]

      19. sub-neg 99.4%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) - x\right)}\right)\right), x\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}\right), x\right) \]
   8. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}, x\right) \]

      2. metadata-eval 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1} \cdot \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right), x\right) \]

      3. *-un-lft-identity 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

      4. neg-sub0 99.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]
   10. Step-by-step derivation

       1. neg-sub0 99.9%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   11. Simplified 99.9%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.00104999999999999994 < x < 1e-3

   1. Initial program 7.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.8%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 3.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 3.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 8.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 8.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 8.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 8.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right) \]

   if 1e-3 < x

   1. Initial program 55.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 55.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 55.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
   4. Step-by-step derivation

      1. +-rgt-identity 99.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   5. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 51.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + -1 \cdot x\right)}, x\right) \]
   3. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| - x\right)}, x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| - x\right)}, x\right) \]

   if -1.26000000000000001 < x < 1.25

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 9.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 9.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 9.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 9.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 9.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right) \]

   if 1.25 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.4)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.4:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.4)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.4)
		tmp = sign(x) * abs(-log((-1.0 / x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.4], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.39999999999999991

   1. Initial program 51.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around -inf 31.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{-1}{x}\right)}, x\right) \]
   3. Step-by-step derivation

      1. mul-1-neg 31.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{-1}{x}\right)}, x\right) \]

   4. Simplified 31.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{-1}{x}\right)}, x\right) \]

   if -3.39999999999999991 < x < 1.25

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 9.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 9.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 9.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 9.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 9.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right) \]

   if 1.25 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (copysign (* 0.3333333333333333 (* 3.0 (log (/ -0.5 x)))) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = copysign((0.3333333333333333 * (3.0 * log((-0.5 / x)))), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.copySign((0.3333333333333333 * (3.0 * Math.log((-0.5 / x)))), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.copysign((0.3333333333333333 * (3.0 * math.log((-0.5 / x)))), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = copysign(Float64(0.3333333333333333 * Float64(3.0 * log(Float64(-0.5 / x)))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = sign(x) * abs((0.3333333333333333 * (3.0 * log((-0.5 / x)))));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[With[{TMP1 = Abs[N[(0.3333333333333333 * N[(3.0 * N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 51.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 27.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 27.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 27.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 27.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 51.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 3.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 3.1%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 3.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around -inf 99.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(\frac{-0.5}{x}\right)}\right), x\right) \]

   if -1.26000000000000001 < x < 1.25

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 9.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 9.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 9.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 9.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 9.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + x}, x\right) \]

   if 1.25 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{-0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 8: 81.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.25) (copysign x x) (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.2)
		tmp = sign(x) * abs(-log((-1.0 / x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.2], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000002

   1. Initial program 51.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around -inf 31.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{-1}{x}\right)}, x\right) \]
   3. Step-by-step derivation

      1. mul-1-neg 31.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{-1}{x}\right)}, x\right) \]

   4. Simplified 31.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{-1}{x}\right)}, x\right) \]

   if -3.2000000000000002 < x < 1.25

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 9.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 9.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 9.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 4.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 9.3%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 9.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 1.25 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6) (copysign x x) (copysign (log (+ x 1.0)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + 1.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + 1.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + 1.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + 1.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 23.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 15.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 15.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 15.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 15.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 2.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 2.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 7.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 7.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 7.2%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 66.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 67.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 1.6000000000000001 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around 0 31.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left|x\right|\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 31.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      2. fabs-sqr 31.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      3. rem-square-sqrt 31.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{x}\right), x\right) \]

   4. Simplified 31.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \]

Alternative 10: 75.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (copysign x x) (copysign (log (+ x x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 23.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 15.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 15.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 15.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 15.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 2.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 2.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 7.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 7.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 7.2%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 66.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 67.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 1.25 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]
   3. Step-by-step derivation

      1. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + x\right), x\right) \]

      2. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + x\right), x\right) \]

      3. rem-square-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + x\right), x\right) \]

   4. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 11: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6) (copysign x x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 23.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. add-cbrt-cube 15.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 15.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 15.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 15.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 23.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. add-sqr-sqrt 2.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      7. fabs-sqr 2.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      8. add-sqr-sqrt 7.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      9. +-commutative 7.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      10. hypot-1-def 7.2%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 66.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 67.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

   if 1.6000000000000001 < x

   1. Initial program 55.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Taylor expanded in x around 0 31.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   3. Step-by-step derivation

      1. log1p-def 31.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. rem-square-sqrt 31.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

      3. fabs-sqr 31.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

      4. rem-square-sqrt 31.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   4. Simplified 31.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 12: 51.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign x x))

C

double code(double x) {
	return copysign(x, x);
}

Java

public static double code(double x) {
	return Math.copySign(x, x);
}

Python

def code(x):
	return math.copysign(x, x)

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

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

Wolfram

code[x_] := N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 31.8%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. add-cbrt-cube 22.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

   2. pow1/3 22.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

   3. log-pow 22.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   4. pow3 22.2%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

   5. log-pow 31.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   6. add-sqr-sqrt 16.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

   7. fabs-sqr 16.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

   8. add-sqr-sqrt 19.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

   9. +-commutative 19.7%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

   10. hypot-1-def 31.3%

       \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

3. Applied egg-rr 31.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

4. Taylor expanded in x around 0 50.2%

   \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right)}\right), x\right) \]

5. Taylor expanded in x around 0 51.0%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]

6. Final simplification 51.0%

   \[\leadsto \mathsf{copysign}\left(x, x\right) \]

Developer target: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t_0\right) + t_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Reproduce

herbie shell --seed 2023242 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))