Hyperbolic arcsine

Percentage Accurate: 17.9% → 100.0%

Time: 10.9s

Alternatives: 10

Speedup: 207.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 17.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\left(x + \mathsf{hypot}\left(1, x\right)\right) \cdot e^{-1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.024)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.02)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* -0.044642857142857144 (* x x))))
         0.16666666666666666))))
     (+ 1.0 (log (* (+ x (hypot 1.0 x)) (exp -1.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.02) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)));
	} else {
		tmp = 1.0 + log(((x + hypot(1.0, x)) * exp(-1.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.02) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)));
	} else {
		tmp = 1.0 + Math.log(((x + Math.hypot(1.0, x)) * Math.exp(-1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.024:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.02:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)))
	else:
		tmp = 1.0 + math.log(((x + math.hypot(1.0, x)) * math.exp(-1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.024)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.02)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64(-0.044642857142857144 * Float64(x * x)))) - 0.16666666666666666))));
	else
		tmp = Float64(1.0 + log(Float64(Float64(x + hypot(1.0, x)) * exp(-1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.024)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.02)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)));
	else
		tmp = 1.0 + log(((x + hypot(1.0, x)) * exp(-1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.024], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.02], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(-0.044642857142857144 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] * N[Exp[-1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   1. Initial program 4.6%

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

      1. sqr-neg 4.6%

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

      2. +-commutative 4.6%

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

      3. sqr-neg 4.6%

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

      4. hypot-1-def 5.8%

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

   3. Simplified 5.8%

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

      1. flip-+ 4.5%

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

      2. frac-2neg 4.5%

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

      3. log-div 4.5%

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

      4. pow2 4.5%

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

      5. hypot-1-def 5.5%

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

      6. hypot-1-def 4.6%

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

      7. add-sqr-sqrt 7.8%

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

      8. +-commutative 7.8%

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

      9. fma-define 7.8%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   6. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 7.8%

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

      2. unpow2 7.8%

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

      3. associate--r+ 52.4%

         \[\leadsto \log \left(-\color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.024 < x < 0.0200000000000000004

   1. Initial program 9.3%

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

      1. sqr-neg 9.3%

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

      2. +-commutative 9.3%

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

      3. sqr-neg 9.3%

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

      4. hypot-1-def 9.3%

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

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   if 0.0200000000000000004 < x

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. +-commutative 52.9%

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

      3. sqr-neg 52.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 98.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1} \]

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. exp-sum 99.9%

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

      3. add-exp-log 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\left(x + \mathsf{hypot}\left(1, x\right)\right) \cdot e^{-1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{x + \mathsf{hypot}\left(1, x\right)}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.024)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.02)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* -0.044642857142857144 (* x x))))
         0.16666666666666666))))
     (+ 1.0 (log (/ (+ x (hypot 1.0 x)) E))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.02) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)));
	} else {
		tmp = 1.0 + log(((x + hypot(1.0, x)) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.02) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)));
	} else {
		tmp = 1.0 + Math.log(((x + Math.hypot(1.0, x)) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.024:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.02:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)))
	else:
		tmp = 1.0 + math.log(((x + math.hypot(1.0, x)) / math.e))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.024)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.02)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64(-0.044642857142857144 * Float64(x * x)))) - 0.16666666666666666))));
	else
		tmp = Float64(1.0 + log(Float64(Float64(x + hypot(1.0, x)) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.024)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.02)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + (-0.044642857142857144 * (x * x)))) - 0.16666666666666666)));
	else
		tmp = 1.0 + log(((x + hypot(1.0, x)) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.024], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.02], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(-0.044642857142857144 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   1. Initial program 4.6%

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

      1. sqr-neg 4.6%

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

      2. +-commutative 4.6%

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

      3. sqr-neg 4.6%

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

      4. hypot-1-def 5.8%

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

   3. Simplified 5.8%

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

      1. flip-+ 4.5%

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

      2. frac-2neg 4.5%

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

      3. log-div 4.5%

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

      4. pow2 4.5%

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

      5. hypot-1-def 5.5%

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

      6. hypot-1-def 4.6%

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

      7. add-sqr-sqrt 7.8%

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

      8. +-commutative 7.8%

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

      9. fma-define 7.8%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   6. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 7.8%

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

      2. unpow2 7.8%

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

      3. associate--r+ 52.4%

         \[\leadsto \log \left(-\color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.024 < x < 0.0200000000000000004

   1. Initial program 9.3%

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

      1. sqr-neg 9.3%

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

      2. +-commutative 9.3%

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

      3. sqr-neg 9.3%

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

      4. hypot-1-def 9.3%

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

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   if 0.0200000000000000004 < x

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. +-commutative 52.9%

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

      3. sqr-neg 52.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 98.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1} \]

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. exp-sum 99.9%

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

      3. add-exp-log 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-commutative 99.9%

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

       2. distribute-lft-in 99.9%

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

   12. Applied egg-rr 99.9%

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

       1. distribute-lft-out 99.9%

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

       2. rem-exp-log 99.9%

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

       3. exp-sum 99.9%

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

       4. +-commutative 99.9%

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

       5. metadata-eval 99.9%

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

       6. sub-neg 99.9%

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

       7. exp-diff 99.9%

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

       8. rem-exp-log 99.9%

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

       9. exp-1-e 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.02:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{x + \mathsf{hypot}\left(1, x\right)}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0082:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{x + \mathsf{hypot}\left(1, x\right)}{e}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0075)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0082)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (+ 1.0 (log (/ (+ x (hypot 1.0 x)) E))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0075) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0082) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = 1.0 + log(((x + hypot(1.0, x)) / ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0075) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0082) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = 1.0 + Math.log(((x + Math.hypot(1.0, x)) / Math.E));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0075:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0082:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = 1.0 + math.log(((x + math.hypot(1.0, x)) / math.e))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0075)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0082)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = Float64(1.0 + log(Float64(Float64(x + hypot(1.0, x)) / exp(1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0075)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0082)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = 1.0 + log(((x + hypot(1.0, x)) / 2.71828182845904523536));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0075], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0082], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074999999999999997

   1. Initial program 4.6%

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

      1. sqr-neg 4.6%

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

      2. +-commutative 4.6%

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

      3. sqr-neg 4.6%

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

      4. hypot-1-def 5.8%

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

   3. Simplified 5.8%

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

      1. flip-+ 4.5%

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

      2. frac-2neg 4.5%

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

      3. log-div 4.5%

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

      4. pow2 4.5%

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

      5. hypot-1-def 5.5%

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

      6. hypot-1-def 4.6%

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

      7. add-sqr-sqrt 7.8%

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

      8. +-commutative 7.8%

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

      9. fma-define 7.8%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   6. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 7.8%

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

      2. unpow2 7.8%

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

      3. associate--r+ 52.4%

         \[\leadsto \log \left(-\color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.0074999999999999997 < x < 0.00820000000000000069

   1. Initial program 9.3%

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

      1. sqr-neg 9.3%

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

      2. +-commutative 9.3%

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

      3. sqr-neg 9.3%

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

      4. hypot-1-def 9.3%

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

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right) \]

   if 0.00820000000000000069 < x

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. +-commutative 52.9%

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

      3. sqr-neg 52.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 98.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 98.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1} \]

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. exp-sum 99.9%

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

      3. add-exp-log 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-commutative 99.9%

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

       2. distribute-lft-in 99.9%

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

   12. Applied egg-rr 99.9%

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

       1. distribute-lft-out 99.9%

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

       2. rem-exp-log 99.9%

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

       3. exp-sum 99.9%

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

       4. +-commutative 99.9%

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

       5. metadata-eval 99.9%

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

       6. sub-neg 99.9%

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

       7. exp-diff 99.9%

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

       8. rem-exp-log 99.9%

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

       9. exp-1-e 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0082:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{x + \mathsf{hypot}\left(1, x\right)}{e}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0075)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0075)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0075) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0075) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0075) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0075) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0075:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0075:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0075)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0075)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0075)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0075)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0075], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0075], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074999999999999997

   1. Initial program 4.6%

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

      1. sqr-neg 4.6%

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

      2. +-commutative 4.6%

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

      3. sqr-neg 4.6%

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

      4. hypot-1-def 5.8%

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

   3. Simplified 5.8%

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

      1. flip-+ 4.5%

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

      2. frac-2neg 4.5%

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

      3. log-div 4.5%

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

      4. pow2 4.5%

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

      5. hypot-1-def 5.5%

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

      6. hypot-1-def 4.6%

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

      7. add-sqr-sqrt 7.8%

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

      8. +-commutative 7.8%

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

      9. fma-define 7.8%

         \[\leadsto \log \left(-\left({x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   6. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\log \left(-\left({x}^{2} - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 7.8%

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

      2. unpow2 7.8%

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

      3. associate--r+ 52.4%

         \[\leadsto \log \left(-\color{blue}{\left(\left({x}^{2} - {x}^{2}\right) - 1\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.0074999999999999997 < x < 0.0074999999999999997

   1. Initial program 9.3%

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

      1. sqr-neg 9.3%

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

      2. +-commutative 9.3%

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

      3. sqr-neg 9.3%

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

      4. hypot-1-def 9.3%

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

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right) \]

   if 0.0074999999999999997 < x

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. +-commutative 52.9%

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

      3. sqr-neg 52.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (log (/ -0.5 x))
   (if (<= x 0.0075)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.0075) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.0075) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.32:
		tmp = math.log((-0.5 / x))
	elif x <= 0.0075:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.0075)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.32)
		tmp = log((-0.5 / x));
	elseif (x <= 0.0075)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0075], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32000000000000006

   1. Initial program 4.6%

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

      1. sqr-neg 4.6%

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

      2. +-commutative 4.6%

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

      3. sqr-neg 4.6%

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

      4. hypot-1-def 5.8%

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

   3. Simplified 5.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.32000000000000006 < x < 0.0074999999999999997

   1. Initial program 9.3%

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

      1. sqr-neg 9.3%

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

      2. +-commutative 9.3%

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

      3. sqr-neg 9.3%

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

      4. hypot-1-def 9.3%

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

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right) \]

   if 0.0074999999999999997 < x

   1. Initial program 52.9%

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

      1. sqr-neg 52.9%

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

      2. +-commutative 52.9%

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

      3. sqr-neg 52.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

Alternative 6: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (log (/ -0.5 x))
   (if (<= x 1.35)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.35) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.32d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.35d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.35) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.32:
		tmp = math.log((-0.5 / x))
	elif x <= 1.35:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.35)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.32)
		tmp = log((-0.5 / x));
	elseif (x <= 1.35)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.35], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32000000000000006

   1. Initial program 4.6%

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

      1. sqr-neg 4.6%

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

      2. +-commutative 4.6%

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

      3. sqr-neg 4.6%

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

      4. hypot-1-def 5.8%

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

   3. Simplified 5.8%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.32000000000000006 < x < 1.3500000000000001

   1. Initial program 10.6%

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

      1. sqr-neg 10.6%

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

      2. +-commutative 10.6%

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

      3. sqr-neg 10.6%

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

      4. hypot-1-def 10.6%

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

   3. Simplified 10.6%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 99.3%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   9. Applied egg-rr 99.3%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right) \]

   if 1.3500000000000001 < x

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. +-commutative 51.6%

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

      3. sqr-neg 51.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

Alternative 7: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35)
   (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
   (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. +-commutative 8.7%

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

      3. sqr-neg 8.7%

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

      4. hypot-1-def 9.1%

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

   3. Simplified 9.1%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 68.4%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   9. Applied egg-rr 68.4%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right) \]

   if 1.3500000000000001 < x

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. +-commutative 51.6%

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

      3. sqr-neg 51.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

Alternative 8: 54.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
   (/ (* x 2.0) (+ x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.6d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. +-commutative 8.7%

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

      3. sqr-neg 8.7%

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

      4. hypot-1-def 9.1%

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

   3. Simplified 9.1%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   7. Applied egg-rr 68.4%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot \color{blue}{\left(x \cdot x\right)}\right) - 0.16666666666666666\right)\right) \]

   9. Applied egg-rr 68.4%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right) \]

   if 1.6000000000000001 < x

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. +-commutative 51.6%

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

      3. sqr-neg 51.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 98.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1} \]

      3. log1p-undefine 98.3%

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

      4. rem-exp-log 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

         \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]

   9. Simplified 5.7%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   10. Step-by-step derivation

       1. flip-- 5.3%

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

       2. pow2 5.3%

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

       3. metadata-eval 5.3%

          \[\leadsto \frac{{\left(x + 1\right)}^{2} - \color{blue}{1}}{\left(x + 1\right) + 1} \]

   11. Applied egg-rr 5.3%

       \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{2} - 1}{\left(x + 1\right) + 1}} \]
   12. Step-by-step derivation

       1. unpow2 5.3%

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

       2. difference-of-sqr-1 5.3%

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

       3. associate-+l+ 5.3%

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

       4. metadata-eval 5.3%

          \[\leadsto \frac{\left(x + \color{blue}{2}\right) \cdot \left(\left(x + 1\right) - 1\right)}{\left(x + 1\right) + 1} \]

       5. sub-neg 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{\left(\left(x + 1\right) + \left(-1\right)\right)}}{\left(x + 1\right) + 1} \]

       6. metadata-eval 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(\left(x + 1\right) + \color{blue}{-1}\right)}{\left(x + 1\right) + 1} \]

       7. associate-+l+ 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{\left(x + \left(1 + -1\right)\right)}}{\left(x + 1\right) + 1} \]

       8. metadata-eval 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

       9. +-rgt-identity 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{x}}{\left(x + 1\right) + 1} \]

       10. associate-+l+ 5.3%

           \[\leadsto \frac{\left(x + 2\right) \cdot x}{\color{blue}{x + \left(1 + 1\right)}} \]

       11. metadata-eval 5.3%

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

   13. Simplified 5.3%

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

   14. Taylor expanded in x around 0 14.5%

       \[\leadsto \frac{\color{blue}{2 \cdot x}}{x + 2} \]
   15. Step-by-step derivation

       1. *-commutative 14.5%

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]

   16. Simplified 14.5%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 54.6% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.8) x (/ (* x 2.0) (+ x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = x;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = x
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = x;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = x
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = x;
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = x;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], x, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. +-commutative 8.7%

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

      3. sqr-neg 8.7%

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

      4. hypot-1-def 9.1%

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

   3. Simplified 9.1%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{x} \]

   if 1.80000000000000004 < x

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. +-commutative 51.6%

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

      3. sqr-neg 51.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 98.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1} \]

      3. log1p-undefine 98.3%

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

      4. rem-exp-log 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

         \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]

   9. Simplified 5.7%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   10. Step-by-step derivation

       1. flip-- 5.3%

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

       2. pow2 5.3%

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

       3. metadata-eval 5.3%

          \[\leadsto \frac{{\left(x + 1\right)}^{2} - \color{blue}{1}}{\left(x + 1\right) + 1} \]

   11. Applied egg-rr 5.3%

       \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{2} - 1}{\left(x + 1\right) + 1}} \]
   12. Step-by-step derivation

       1. unpow2 5.3%

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

       2. difference-of-sqr-1 5.3%

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

       3. associate-+l+ 5.3%

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

       4. metadata-eval 5.3%

          \[\leadsto \frac{\left(x + \color{blue}{2}\right) \cdot \left(\left(x + 1\right) - 1\right)}{\left(x + 1\right) + 1} \]

       5. sub-neg 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{\left(\left(x + 1\right) + \left(-1\right)\right)}}{\left(x + 1\right) + 1} \]

       6. metadata-eval 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(\left(x + 1\right) + \color{blue}{-1}\right)}{\left(x + 1\right) + 1} \]

       7. associate-+l+ 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{\left(x + \left(1 + -1\right)\right)}}{\left(x + 1\right) + 1} \]

       8. metadata-eval 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

       9. +-rgt-identity 5.3%

          \[\leadsto \frac{\left(x + 2\right) \cdot \color{blue}{x}}{\left(x + 1\right) + 1} \]

       10. associate-+l+ 5.3%

           \[\leadsto \frac{\left(x + 2\right) \cdot x}{\color{blue}{x + \left(1 + 1\right)}} \]

       11. metadata-eval 5.3%

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

   13. Simplified 5.3%

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

   14. Taylor expanded in x around 0 14.5%

       \[\leadsto \frac{\color{blue}{2 \cdot x}}{x + 2} \]
   15. Step-by-step derivation

       1. *-commutative 14.5%

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]

   16. Simplified 14.5%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 19.1%

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

   1. sqr-neg 19.1%

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

   2. +-commutative 19.1%

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

   3. sqr-neg 19.1%

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

   4. hypot-1-def 31.1%

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

3. Simplified 31.1%

   \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.9%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 30.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot x + 1}\\ \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) 1.0))))
   (if (< x 0.0) (log (/ -1.0 (- x t_0))) (log (+ x t_0)))))

C

double code(double x) {
	double t_0 = sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = log((-1.0 / (x - t_0)));
	} else {
		tmp = log((x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + 1.0d0))
    if (x < 0.0d0) then
        tmp = log(((-1.0d0) / (x - t_0)))
    else
        tmp = log((x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = Math.log((-1.0 / (x - t_0)));
	} else {
		tmp = Math.log((x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(((x * x) + 1.0))
	tmp = 0
	if x < 0.0:
		tmp = math.log((-1.0 / (x - t_0)))
	else:
		tmp = math.log((x + t_0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(Float64(x * x) + 1.0))
	tmp = 0.0
	if (x < 0.0)
		tmp = log(Float64(-1.0 / Float64(x - t_0)));
	else
		tmp = log(Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(((x * x) + 1.0));
	tmp = 0.0;
	if (x < 0.0)
		tmp = log((-1.0 / (x - t_0)));
	else
		tmp = log((x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[Less[x, 0.0], N[Log[N[(-1.0 / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + t$95$0), $MachinePrecision]], $MachinePrecision]]]

Reproduce

herbie shell --seed 2024145 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

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