Rust f64::asinh

Percentage Accurate: 29.8% → 98.9%

Time: 4.5s

Alternatives: 9

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: 29.8% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 10^{-9}:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 1e-9)
       (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) 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 <= -5.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-9) {
		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 t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-9) {
		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):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 1e-9:
		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)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 1e-9)
		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)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -5.0)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 1e-9)
		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_] := 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, -5.0], 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, 1e-9], 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 (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -5

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. log-prod 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 5.0%

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

   5. Applied egg-rr 5.0%

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

      1. +-lft-identity 5.0%

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

   7. Simplified 5.0%

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

      1. flip-+ 3.8%

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

      2. frac-2neg 3.8%

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

      3. log-div 3.8%

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

      4. hypot-udef 4.2%

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

      5. hypot-udef 3.8%

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

      6. add-sqr-sqrt 4.2%

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

      7. metadata-eval 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-def 4.2%

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

   9. Applied egg-rr 4.2%

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

       1. neg-mul-1 4.2%

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

       2. fma-udef 4.2%

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

       3. associate--r+ 47.9%

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

       4. +-inverses 100.0%

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

       5. metadata-eval 100.0%

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

       6. metadata-eval 100.0%

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

       7. metadata-eval 100.0%

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

       8. neg-sub0 100.0%

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

       9. neg-sub0 100.0%

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

       10. associate--r- 100.0%

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

       11. neg-sub0 100.0%

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

       12. +-commutative 100.0%

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

       13. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 1.00000000000000006e-9

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

      1. *-un-lft-identity 6.7%

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

      2. log-prod 6.7%

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

      3. metadata-eval 6.7%

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

      4. *-un-lft-identity 6.7%

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

      5. *-un-lft-identity 6.7%

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

      6. add-sqr-sqrt 2.9%

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

      7. fabs-sqr 2.9%

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

      8. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. +-lft-identity 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000006e-9 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + x\right), x\right) \]

      3. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + x\right), x\right) \]

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-9}:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (copysign (- (log (* x -2.0))) 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 <= -1.3) {
		tmp = copysign(-log((x * -2.0)), 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 <= -1.3) {
		tmp = Math.copySign(-Math.log((x * -2.0)), 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 <= -1.3:
		tmp = math.copysign(-math.log((x * -2.0)), 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 <= -1.3)
		tmp = copysign(Float64(-log(Float64(x * -2.0))), 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 <= -1.3)
		tmp = sign(x) * abs(-log((x * -2.0)));
	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, -1.3], N[With[{TMP1 = Abs[(-N[Log[N[(x * -2.0), $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 < -1.30000000000000004

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right)}, x\right) \]

      2. unpow1 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{1}}\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      3. sqr-pow 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      4. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      5. sqr-pow 4.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{1}} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      6. unpow1 4.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      7. associate-+r- 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]

      8. mul-1-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(x + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      9. sub-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(x - x\right)} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      10. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)}, x\right) \]

      12. associate-*r/ 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right), x\right) \]

      13. metadata-eval 98.9%

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

      14. distribute-neg-frac 98.9%

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

      15. metadata-eval 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]
   7. Step-by-step derivation

      1. clear-num 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{x}{-0.5}}\right)}, x\right) \]

      2. log-rec 98.9%

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

      3. div-inv 98.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}, x\right) \]

      4. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-2}\right), x\right) \]

   8. Applied egg-rr 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(x \cdot -2\right)}, x\right) \]

   if -1.30000000000000004 < x < 1.25

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

      1. *-un-lft-identity 6.7%

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

      2. log-prod 6.7%

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

      3. metadata-eval 6.7%

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

      4. *-un-lft-identity 6.7%

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

      5. *-un-lft-identity 6.7%

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

      6. add-sqr-sqrt 2.9%

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

      7. fabs-sqr 2.9%

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

      8. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. +-lft-identity 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + x\right), x\right) \]

      3. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + x\right), x\right) \]

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (copysign (- (log (- (* x -2.0) (/ 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 <= -0.96) {
		tmp = copysign(-log(((x * -2.0) - (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 <= -0.96) {
		tmp = Math.copySign(-Math.log(((x * -2.0) - (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 <= -0.96:
		tmp = math.copysign(-math.log(((x * -2.0) - (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 <= -0.96)
		tmp = copysign(Float64(-log(Float64(Float64(x * -2.0) - 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 <= -0.96)
		tmp = sign(x) * abs(-log(((x * -2.0) - (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, -0.96], N[With[{TMP1 = Abs[(-N[Log[N[(N[(x * -2.0), $MachinePrecision] - N[(0.5 / x), $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 < -0.95999999999999996

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. log-prod 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 5.0%

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

   5. Applied egg-rr 5.0%

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

      1. +-lft-identity 5.0%

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

   7. Simplified 5.0%

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

      1. flip-+ 3.8%

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

      2. frac-2neg 3.8%

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

      3. log-div 3.8%

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

      4. hypot-udef 4.2%

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

      5. hypot-udef 3.8%

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

      6. add-sqr-sqrt 4.2%

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

      7. metadata-eval 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-def 4.2%

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

   9. Applied egg-rr 4.2%

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

       1. neg-mul-1 4.2%

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

       2. fma-udef 4.2%

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

       3. associate--r+ 47.9%

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

       4. +-inverses 100.0%

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

       5. metadata-eval 100.0%

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

       6. metadata-eval 100.0%

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

       7. metadata-eval 100.0%

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

       8. neg-sub0 100.0%

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

       9. neg-sub0 100.0%

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

       10. associate--r- 100.0%

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

       11. neg-sub0 100.0%

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

       12. +-commutative 100.0%

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

       13. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around -inf 99.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
   13. Step-by-step derivation

       1. *-commutative 99.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

       2. associate-*r/ 99.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right), x\right) \]

       3. metadata-eval 99.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{\color{blue}{0.5}}{x}\right), x\right) \]

   14. Simplified 99.8%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2 - \frac{0.5}{x}\right)}, x\right) \]

   if -0.95999999999999996 < x < 1.25

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

      1. *-un-lft-identity 6.7%

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

      2. log-prod 6.7%

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

      3. metadata-eval 6.7%

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

      4. *-un-lft-identity 6.7%

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

      5. *-un-lft-identity 6.7%

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

      6. add-sqr-sqrt 2.9%

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

      7. fabs-sqr 2.9%

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

      8. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. +-lft-identity 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + x\right), x\right) \]

      3. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + x\right), x\right) \]

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. 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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 4: 82.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (log (- 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(-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(-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(-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(log(Float64(-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(-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[(-x)], $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 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

   6. Simplified 31.4%

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

   if -3.2000000000000002 < x < 1.25

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in x around 0 6.5%

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

      1. unpow1 6.5%

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

      2. sqr-pow 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

      3. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

      4. sqr-pow 6.6%

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

      5. unpow1 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 1.25 < x

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + x\right), x\right) \]

      3. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + x\right), x\right) \]

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. 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 80.4%

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

Alternative 5: 99.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (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(log((-0.5 / 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 <= -1.25) {
		tmp = Math.copySign(Math.log((-0.5 / 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 <= -1.25:
		tmp = math.copysign(math.log((-0.5 / 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 <= -1.25)
		tmp = copysign(log(Float64(-0.5 / 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 <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / 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, -1.25], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / 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 < -1.25

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right)}, x\right) \]

      2. unpow1 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{1}}\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      3. sqr-pow 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      4. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      5. sqr-pow 4.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{1}} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      6. unpow1 4.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      7. associate-+r- 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]

      8. mul-1-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(x + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      9. sub-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(x - x\right)} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      10. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)}, x\right) \]

      12. associate-*r/ 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right), x\right) \]

      13. metadata-eval 98.9%

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

      14. distribute-neg-frac 98.9%

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

      15. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

   if -1.25 < x < 1.25

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in x around 0 6.5%

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

      1. unpow1 6.5%

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

      2. sqr-pow 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

      3. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

      4. sqr-pow 6.6%

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

      5. unpow1 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 1.25 < x

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + x\right), x\right) \]

      3. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + x\right), x\right) \]

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. 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%

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

Alternative 6: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (- (log (* x -2.0))) x)
   (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(-log((x * -2.0)), 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 <= -1.25) {
		tmp = Math.copySign(-Math.log((x * -2.0)), 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 <= -1.25:
		tmp = math.copysign(-math.log((x * -2.0)), 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 <= -1.25)
		tmp = copysign(Float64(-log(Float64(x * -2.0))), 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 <= -1.25)
		tmp = sign(x) * abs(-log((x * -2.0)));
	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, -1.25], N[With[{TMP1 = Abs[(-N[Log[N[(x * -2.0), $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 < -1.25

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right)}, x\right) \]

      2. unpow1 99.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{1}}\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      3. sqr-pow 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      4. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      5. sqr-pow 4.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{1}} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      6. unpow1 4.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]

      7. associate-+r- 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]

      8. mul-1-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(x + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      9. sub-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(x - x\right)} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      10. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)}, x\right) \]

      12. associate-*r/ 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right), x\right) \]

      13. metadata-eval 98.9%

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

      14. distribute-neg-frac 98.9%

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

      15. metadata-eval 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]
   7. Step-by-step derivation

      1. clear-num 98.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{x}{-0.5}}\right)}, x\right) \]

      2. log-rec 98.9%

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

      3. div-inv 98.9%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}, x\right) \]

      4. metadata-eval 98.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot \color{blue}{-2}\right), x\right) \]

   8. Applied egg-rr 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(x \cdot -2\right)}, x\right) \]

   if -1.25 < x < 1.25

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in x around 0 6.5%

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

      1. unpow1 6.5%

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

      2. sqr-pow 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

      3. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

      4. sqr-pow 6.6%

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

      5. unpow1 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in x around 0 99.9%

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

   if 1.25 < x

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| + x\right), x\right) \]

      3. fabs-sqr 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}} + x\right), x\right) \]

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.5) (copysign (log (- x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(log(-x), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = Math.copySign(Math.log(-x), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.5:
		tmp = math.copysign(math.log(-x), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.5)
		tmp = copysign(log(Float64(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.5], N[With[{TMP1 = Abs[N[Log[(-x)], $MachinePrecision]], 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 < -0.5

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

   6. Simplified 31.4%

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

   if -0.5 < x

   1. Initial program 21.4%

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

      1. +-commutative 21.4%

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

      2. hypot-1-def 36.2%

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

   3. Simplified 36.2%

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

   4. Taylor expanded in x around 0 14.4%

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

      1. log1p-def 77.7%

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

      2. unpow1 77.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{1}}\right|\right), x\right) \]

      3. sqr-pow 41.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

      4. fabs-sqr 41.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

      5. sqr-pow 77.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{1}}\right), x\right) \]

      6. unpow1 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 8: 59.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (copysign x x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], 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 < 1.55000000000000004

   1. Initial program 22.5%

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

      1. +-commutative 22.5%

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

      2. hypot-1-def 41.1%

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

   3. Simplified 41.1%

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

   4. Taylor expanded in x around 0 15.7%

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

      1. unpow1 15.7%

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

      2. sqr-pow 1.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

      3. fabs-sqr 1.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

      4. sqr-pow 4.1%

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

      5. unpow1 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in x around 0 65.1%

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

   if 1.55000000000000004 < x

   1. Initial program 53.2%

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

      1. +-commutative 53.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.5%

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

      1. log1p-def 31.5%

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

      2. unpow1 31.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{1}}\right|\right), x\right) \]

      3. sqr-pow 31.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

      4. fabs-sqr 31.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

      5. sqr-pow 31.5%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{1}}\right), x\right) \]

      6. unpow1 31.5%

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

   6. Simplified 31.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 9: 52.8% 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 29.4%

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

   1. +-commutative 29.4%

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

   2. hypot-1-def 54.4%

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

3. Simplified 54.4%

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

4. Taylor expanded in x around 0 19.3%

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

   1. unpow1 19.3%

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

   2. sqr-pow 8.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right), x\right) \]

   3. fabs-sqr 8.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right), x\right) \]

   4. sqr-pow 10.3%

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

   5. unpow1 10.3%

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

6. Simplified 10.3%

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

7. Taylor expanded in x around 0 51.5%

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

8. Final simplification 51.5%

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

Developer target: 100.0% 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 2023178 
(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))