Hyperbolic arcsine

Percentage Accurate: 17.9% → 99.9%

Time: 11.7s

Alternatives: 17

Speedup: 18.8×

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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + x \cdot \left(1 - \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.05)
   (- (log1p (+ (hypot 1.0 x) (- -1.0 x))))
   (if (<= x 0.022)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
         0.16666666666666666))))
     (log1p (+ (hypot 1.0 x) (* x (- 1.0 (/ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.05) {
		tmp = -log1p((hypot(1.0, x) + (-1.0 - x)));
	} else if (x <= 0.022) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	} else {
		tmp = log1p((hypot(1.0, x) + (x * (1.0 - (1.0 / x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.05) {
		tmp = -Math.log1p((Math.hypot(1.0, x) + (-1.0 - x)));
	} else if (x <= 0.022) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (Math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	} else {
		tmp = Math.log1p((Math.hypot(1.0, x) + (x * (1.0 - (1.0 / x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.05:
		tmp = -math.log1p((math.hypot(1.0, x) + (-1.0 - x)))
	elif x <= 0.022:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))
	else:
		tmp = math.log1p((math.hypot(1.0, x) + (x * (1.0 - (1.0 / x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.05)
		tmp = Float64(-log1p(Float64(hypot(1.0, x) + Float64(-1.0 - x))));
	elseif (x <= 0.022)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666))));
	else
		tmp = log1p(Float64(hypot(1.0, x) + Float64(x * Float64(1.0 - Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.05], (-N[Log[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.022], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(x * N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.050000000000000003

   1. Initial program 5.1%

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

      1. flip-+ 4.8%

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

      2. frac-2neg 4.8%

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

      3. log-div 4.8%

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

      4. add-sqr-sqrt 4.8%

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

      5. pow2 4.8%

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

      6. fma-define 4.8%

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

      7. +-commutative 4.8%

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

      8. hypot-1-def 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. fma-undefine 4.8%

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

      2. unpow2 4.8%

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

      3. associate--r+ 49.0%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. neg-sub0 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. neg-sub0 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.9%

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

      3. sqr-neg 99.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.5%

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

      6. log1p-expm1-u 0.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 99.9%

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

      9. sqr-neg 99.9%

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

      10. sqrt-unprod 99.1%

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

      11. add-sqr-sqrt 99.9%

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

      12. expm1-undefine 99.9%

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

      13. add-exp-log 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) - x\right) - 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l- 100.0%

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

   10. Simplified 100.0%

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

   if -0.050000000000000003 < x < 0.021999999999999999

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

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

   if 0.021999999999999999 < x

   1. Initial program 57.7%

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

      1. add-sqr-sqrt 57.7%

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

      2. pow2 57.7%

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

      3. log-pow 57.8%

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

      4. +-commutative 57.8%

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

      5. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-undefine 99.9%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-undefine 99.9%

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

      4. *-commutative 99.9%

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

      5. exp-to-pow 99.9%

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

      6. pow2 99.9%

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

      7. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. log1p-define 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \color{blue}{x \cdot \left(1 - \frac{1}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + x \cdot \left(1 - \frac{1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.0138:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot \left(1 + x \cdot \left(0.5 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0625 - 0.125\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + x \cdot \left(1 - \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.014)
   (- (log1p (+ (hypot 1.0 x) (- -1.0 x))))
   (if (<= x 0.0138)
     (log1p
      (*
       x
       (+ 1.0 (* x (+ 0.5 (* (pow x 2.0) (- (* (pow x 2.0) 0.0625) 0.125)))))))
     (log1p (+ (hypot 1.0 x) (* x (- 1.0 (/ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.014) {
		tmp = -log1p((hypot(1.0, x) + (-1.0 - x)));
	} else if (x <= 0.0138) {
		tmp = log1p((x * (1.0 + (x * (0.5 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.0625) - 0.125)))))));
	} else {
		tmp = log1p((hypot(1.0, x) + (x * (1.0 - (1.0 / x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.014) {
		tmp = -Math.log1p((Math.hypot(1.0, x) + (-1.0 - x)));
	} else if (x <= 0.0138) {
		tmp = Math.log1p((x * (1.0 + (x * (0.5 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.0625) - 0.125)))))));
	} else {
		tmp = Math.log1p((Math.hypot(1.0, x) + (x * (1.0 - (1.0 / x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.014:
		tmp = -math.log1p((math.hypot(1.0, x) + (-1.0 - x)))
	elif x <= 0.0138:
		tmp = math.log1p((x * (1.0 + (x * (0.5 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.0625) - 0.125)))))))
	else:
		tmp = math.log1p((math.hypot(1.0, x) + (x * (1.0 - (1.0 / x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.014)
		tmp = Float64(-log1p(Float64(hypot(1.0, x) + Float64(-1.0 - x))));
	elseif (x <= 0.0138)
		tmp = log1p(Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.0625) - 0.125)))))));
	else
		tmp = log1p(Float64(hypot(1.0, x) + Float64(x * Float64(1.0 - Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.014], (-N[Log[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0138], N[Log[1 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.0625), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(x * N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0140000000000000003

   1. Initial program 5.1%

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

      1. flip-+ 4.8%

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

      2. frac-2neg 4.8%

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

      3. log-div 4.8%

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

      4. add-sqr-sqrt 4.8%

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

      5. pow2 4.8%

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

      6. fma-define 4.8%

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

      7. +-commutative 4.8%

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

      8. hypot-1-def 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. fma-undefine 4.8%

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

      2. unpow2 4.8%

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

      3. associate--r+ 49.0%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. neg-sub0 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. neg-sub0 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.9%

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

      3. sqr-neg 99.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.5%

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

      6. log1p-expm1-u 0.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 99.9%

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

      9. sqr-neg 99.9%

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

      10. sqrt-unprod 99.1%

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

      11. add-sqr-sqrt 99.9%

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

      12. expm1-undefine 99.9%

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

      13. add-exp-log 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) - x\right) - 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l- 100.0%

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

   10. Simplified 100.0%

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

   if -0.0140000000000000003 < x < 0.0138

   1. Initial program 9.9%

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

      1. add-sqr-sqrt 9.9%

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

      2. pow2 9.9%

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

      3. log-pow 9.9%

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

      4. +-commutative 9.9%

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

      5. hypot-1-def 9.8%

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

   4. Applied egg-rr 9.8%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 9.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-undefine 9.8%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-undefine 9.8%

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

      4. *-commutative 9.8%

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

      5. exp-to-pow 9.8%

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

      6. pow2 9.8%

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

      7. add-sqr-sqrt 9.9%

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

   6. Applied egg-rr 9.9%

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

      1. log1p-define 9.9%

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

      2. sub-neg 9.9%

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

      3. +-commutative 9.9%

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

      4. metadata-eval 9.9%

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

      5. associate-+l+ 9.8%

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

   8. Simplified 9.8%

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

   9. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + {x}^{2} \cdot \left(0.0625 \cdot {x}^{2} - 0.125\right)\right)\right)}\right) \]

   if 0.0138 < x

   1. Initial program 57.7%

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

      1. add-sqr-sqrt 57.7%

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

      2. pow2 57.7%

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

      3. log-pow 57.8%

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

      4. +-commutative 57.8%

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

      5. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-undefine 99.9%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-undefine 99.9%

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

      4. *-commutative 99.9%

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

      5. exp-to-pow 99.9%

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

      6. pow2 99.9%

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

      7. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. log1p-define 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \color{blue}{x \cdot \left(1 - \frac{1}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.014:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.0138:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot \left(1 + x \cdot \left(0.5 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0625 - 0.125\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + x \cdot \left(1 - \frac{1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.007:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + x \cdot \left(1 - \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0072)
   (- (log1p (+ (hypot 1.0 x) (- -1.0 x))))
   (if (<= x 0.007)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log1p (+ (hypot 1.0 x) (* x (- 1.0 (/ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0072) {
		tmp = -log1p((hypot(1.0, x) + (-1.0 - x)));
	} else if (x <= 0.007) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log1p((hypot(1.0, x) + (x * (1.0 - (1.0 / x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0072) {
		tmp = -Math.log1p((Math.hypot(1.0, x) + (-1.0 - x)));
	} else if (x <= 0.007) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log1p((Math.hypot(1.0, x) + (x * (1.0 - (1.0 / x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0072:
		tmp = -math.log1p((math.hypot(1.0, x) + (-1.0 - x)))
	elif x <= 0.007:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log1p((math.hypot(1.0, x) + (x * (1.0 - (1.0 / x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0072)
		tmp = Float64(-log1p(Float64(hypot(1.0, x) + Float64(-1.0 - x))));
	elseif (x <= 0.007)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log1p(Float64(hypot(1.0, x) + Float64(x * Float64(1.0 - Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.0072], (-N[Log[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.007], 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[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(x * N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0071999999999999998

   1. Initial program 5.1%

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

      1. flip-+ 4.8%

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

      2. frac-2neg 4.8%

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

      3. log-div 4.8%

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

      4. add-sqr-sqrt 4.8%

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

      5. pow2 4.8%

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

      6. fma-define 4.8%

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

      7. +-commutative 4.8%

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

      8. hypot-1-def 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. fma-undefine 4.8%

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

      2. unpow2 4.8%

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

      3. associate--r+ 49.0%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. neg-sub0 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. neg-sub0 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.9%

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

      3. sqr-neg 99.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.5%

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

      6. log1p-expm1-u 0.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 99.9%

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

      9. sqr-neg 99.9%

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

      10. sqrt-unprod 99.1%

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

      11. add-sqr-sqrt 99.9%

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

      12. expm1-undefine 99.9%

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

      13. add-exp-log 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) - x\right) - 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l- 100.0%

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

   10. Simplified 100.0%

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

   if -0.0071999999999999998 < x < 0.00700000000000000015

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow2 99.9%

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

   7. 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 0.00700000000000000015 < x

   1. Initial program 57.7%

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

      1. add-sqr-sqrt 57.7%

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

      2. pow2 57.7%

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

      3. log-pow 57.8%

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

      4. +-commutative 57.8%

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

      5. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-undefine 99.9%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-undefine 99.9%

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

      4. *-commutative 99.9%

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

      5. exp-to-pow 99.9%

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

      6. pow2 99.9%

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

      7. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. log1p-define 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-+l+ 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \color{blue}{x \cdot \left(1 - \frac{1}{x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.007:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + x \cdot \left(1 - \frac{1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0072)
   (- (log1p (+ (hypot 1.0 x) (- -1.0 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 <= -0.0072) {
		tmp = -log1p((hypot(1.0, x) + (-1.0 - 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 <= -0.0072) {
		tmp = -Math.log1p((Math.hypot(1.0, x) + (-1.0 - 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 <= -0.0072:
		tmp = -math.log1p((math.hypot(1.0, x) + (-1.0 - 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 <= -0.0072)
		tmp = Float64(-log1p(Float64(hypot(1.0, x) + Float64(-1.0 - 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

Wolfram

code[x_] := If[LessEqual[x, -0.0072], (-N[Log[1 + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $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 < -0.0071999999999999998

   1. Initial program 5.1%

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

      1. flip-+ 4.8%

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

      2. frac-2neg 4.8%

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

      3. log-div 4.8%

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

      4. add-sqr-sqrt 4.8%

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

      5. pow2 4.8%

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

      6. fma-define 4.8%

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

      7. +-commutative 4.8%

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

      8. hypot-1-def 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. fma-undefine 4.8%

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

      2. unpow2 4.8%

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

      3. associate--r+ 49.0%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. neg-sub0 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. neg-sub0 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.9%

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

      3. sqr-neg 99.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 1.5%

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

      6. log1p-expm1-u 0.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 99.9%

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

      9. sqr-neg 99.9%

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

      10. sqrt-unprod 99.1%

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

      11. add-sqr-sqrt 99.9%

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

      12. expm1-undefine 99.9%

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

      13. add-exp-log 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) - x\right) - 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-+l- 100.0%

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

   10. Simplified 100.0%

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

   if -0.0071999999999999998 < x < 0.0074999999999999997

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow2 99.9%

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

   7. 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 57.7%

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

      1. sqr-neg 57.7%

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

      2. +-commutative 57.7%

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

      3. sqr-neg 57.7%

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

      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. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;-\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right)\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0068:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0068)
   (- (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 <= -0.0068) {
		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 <= -0.0068) {
		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 <= -0.0068:
		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 <= -0.0068)
		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 <= -0.0068)
		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, -0.0068], (-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 < -0.00679999999999999962

   1. Initial program 5.1%

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

      1. flip-+ 4.8%

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

      2. frac-2neg 4.8%

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

      3. log-div 4.8%

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

      4. add-sqr-sqrt 4.8%

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

      5. pow2 4.8%

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

      6. fma-define 4.8%

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

      7. +-commutative 4.8%

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

      8. hypot-1-def 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. fma-undefine 4.8%

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

      2. unpow2 4.8%

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

      3. associate--r+ 49.0%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. neg-sub0 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

      13. neg-sub0 99.9%

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

   6. Simplified 99.9%

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

   if -0.00679999999999999962 < x < 0.0074999999999999997

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow2 99.9%

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

   7. 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 57.7%

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

      1. sqr-neg 57.7%

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

      2. +-commutative 57.7%

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

      3. sqr-neg 57.7%

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

      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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (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 <= -1.3) {
		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 <= -1.3) {
		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 <= -1.3:
		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 <= -1.3)
		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 <= -1.3)
		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, -1.3], 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 < -1.30000000000000004

   1. Initial program 3.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.7%

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

   if -1.30000000000000004 < x < 0.0074999999999999997

   1. Initial program 10.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow2 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 0.0074999999999999997 < x

   1. Initial program 57.7%

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

      1. sqr-neg 57.7%

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

      2. +-commutative 57.7%

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

      3. sqr-neg 57.7%

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

      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 7: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.3)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.3d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.3:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.3)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], 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 + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 3.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.7%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 11.3%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.2%

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

      1. unpow2 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

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

   if 1.30000000000000004 < x

   1. Initial program 57.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

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

Alternative 8: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-42.666666666666664\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   -42.666666666666664
   (if (<= x 1.3)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 1.3) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = -42.666666666666664d0
    else if (x <= 1.3d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 1.3) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = -42.666666666666664
	elif x <= 1.3:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 1.3)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 1.3)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], -42.666666666666664, If[LessEqual[x, 1.3], 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 + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000009

   1. Initial program 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. log-pow 1.8%

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

      4. +-commutative 1.8%

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

      5. hypot-1-def 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 1.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 17.2%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -3.60000000000000009 < x < 1.30000000000000004

   1. Initial program 12.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. unpow2 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. unpow2 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 1.30000000000000004 < x

   1. Initial program 57.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

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

Alternative 9: 61.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-42.666666666666664\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   -42.666666666666664
   (if (<= x 1.55)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 1.55) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = -42.666666666666664d0
    else if (x <= 1.55d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = log((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 1.55) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = -42.666666666666664
	elif x <= 1.55:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 1.55)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 1.55)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], -42.666666666666664, If[LessEqual[x, 1.55], 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 + 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000009

   1. Initial program 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. log-pow 1.8%

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

      4. +-commutative 1.8%

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

      5. hypot-1-def 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 1.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 17.2%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -3.60000000000000009 < x < 1.55000000000000004

   1. Initial program 12.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. unpow2 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. unpow2 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 1.55000000000000004 < x

   1. Initial program 57.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 31.2%

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

Alternative 10: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-42.666666666666664\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   -42.666666666666664
   (if (<= x 1.55)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log1p x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 1.55) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log1p(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 1.55) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log1p(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = -42.666666666666664
	elif x <= 1.55:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log1p(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 1.55)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log1p(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], -42.666666666666664, If[LessEqual[x, 1.55], 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[1 + x], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000009

   1. Initial program 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. log-pow 1.8%

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

      4. +-commutative 1.8%

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

      5. hypot-1-def 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 1.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 17.2%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -3.60000000000000009 < x < 1.55000000000000004

   1. Initial program 12.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. unpow2 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. unpow2 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 1.55000000000000004 < x

   1. Initial program 57.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 31.2%

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

      1. *-un-lft-identity 31.2%

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

      2. log-prod 31.2%

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

      3. metadata-eval 31.2%

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

      4. +-commutative 31.2%

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

      5. log1p-define 31.2%

         \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(x\right)} \]

   5. Applied egg-rr 31.2%

      \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(x\right)} \]
   6. Step-by-step derivation

      1. +-lft-identity 31.2%

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

   7. Simplified 31.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-42.666666666666664\\ \mathbf{elif}\;x \leq 3.6:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;43.666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   -42.666666666666664
   (if (<= x 3.6)
     (* x (+ 1.0 (* (* x x) (- (* 0.075 (* x x)) 0.16666666666666666))))
     43.666666666666664)))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 3.6) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = -42.666666666666664d0
    else if (x <= 3.6d0) then
        tmp = x * (1.0d0 + ((x * x) * ((0.075d0 * (x * x)) - 0.16666666666666666d0)))
    else
        tmp = 43.666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -42.666666666666664;
	} else if (x <= 3.6) {
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = -42.666666666666664
	elif x <= 3.6:
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = 43.666666666666664
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 3.6)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = 43.666666666666664;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = -42.666666666666664;
	elseif (x <= 3.6)
		tmp = x * (1.0 + ((x * x) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = 43.666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], -42.666666666666664, If[LessEqual[x, 3.6], 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], 43.666666666666664]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000009

   1. Initial program 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. log-pow 1.8%

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

      4. +-commutative 1.8%

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

      5. hypot-1-def 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 1.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 17.2%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -3.60000000000000009 < x < 3.60000000000000009

   1. Initial program 12.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. unpow2 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. unpow2 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 3.60000000000000009 < x

   1. Initial program 57.2%

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

      1. add-sqr-sqrt 57.2%

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

      2. pow2 57.2%

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

      3. log-pow 57.2%

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

      4. +-commutative 57.2%

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

      5. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 11.2%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 11.2%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 11.2%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around inf 17.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{1}{x}\right)\right) + \left(8 \cdot \frac{1}{x} + 64 \cdot \frac{1}{{x}^{4}}\right)\right) - \frac{21.333333333333332}{{x}^{3}}} \]

   10. Simplified 16.8%

       \[\leadsto \color{blue}{43.666666666666664} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.0% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;-42.666666666666664\\ \mathbf{elif}\;x \leq 2.4:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;43.666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.4)
   -42.666666666666664
   (if (<= x 2.4)
     (* x (+ 1.0 (* (* x x) -0.16666666666666666)))
     43.666666666666664)))

C

double code(double x) {
	double tmp;
	if (x <= -2.4) {
		tmp = -42.666666666666664;
	} else if (x <= 2.4) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.4d0)) then
        tmp = -42.666666666666664d0
    else if (x <= 2.4d0) then
        tmp = x * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = 43.666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.4) {
		tmp = -42.666666666666664;
	} else if (x <= 2.4) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.4:
		tmp = -42.666666666666664
	elif x <= 2.4:
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = 43.666666666666664
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.4)
		tmp = -42.666666666666664;
	elseif (x <= 2.4)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = 43.666666666666664;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.4)
		tmp = -42.666666666666664;
	elseif (x <= 2.4)
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = 43.666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.4], -42.666666666666664, If[LessEqual[x, 2.4], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 43.666666666666664]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999991

   1. Initial program 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. log-pow 1.8%

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

      4. +-commutative 1.8%

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

      5. hypot-1-def 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 1.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 17.2%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -2.39999999999999991 < x < 2.39999999999999991

   1. Initial program 12.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

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

      1. unpow2 98.5%

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

   5. Applied egg-rr 98.5%

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

      1. unpow2 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in x around 0 98.0%

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

   if 2.39999999999999991 < x

   1. Initial program 57.2%

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

      1. add-sqr-sqrt 57.2%

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

      2. pow2 57.2%

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

      3. log-pow 57.2%

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

      4. +-commutative 57.2%

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

      5. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 11.2%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 11.2%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 11.2%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around inf 17.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{1}{x}\right)\right) + \left(8 \cdot \frac{1}{x} + 64 \cdot \frac{1}{{x}^{4}}\right)\right) - \frac{21.333333333333332}{{x}^{3}}} \]

   10. Simplified 16.8%

       \[\leadsto \color{blue}{43.666666666666664} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.7% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -44:\\ \;\;\;\;-42.666666666666664\\ \mathbf{elif}\;x \leq 43:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;43.666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -44.0) -42.666666666666664 (if (<= x 43.0) x 43.666666666666664)))

C

double code(double x) {
	double tmp;
	if (x <= -44.0) {
		tmp = -42.666666666666664;
	} else if (x <= 43.0) {
		tmp = x;
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-44.0d0)) then
        tmp = -42.666666666666664d0
    else if (x <= 43.0d0) then
        tmp = x
    else
        tmp = 43.666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -44.0) {
		tmp = -42.666666666666664;
	} else if (x <= 43.0) {
		tmp = x;
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -44.0:
		tmp = -42.666666666666664
	elif x <= 43.0:
		tmp = x
	else:
		tmp = 43.666666666666664
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -44.0)
		tmp = -42.666666666666664;
	elseif (x <= 43.0)
		tmp = x;
	else
		tmp = 43.666666666666664;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -44.0)
		tmp = -42.666666666666664;
	elseif (x <= 43.0)
		tmp = x;
	else
		tmp = 43.666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -44.0], -42.666666666666664, If[LessEqual[x, 43.0], x, 43.666666666666664]]

Derivation

1. Split input into 3 regimes

2. if x < -44

   1. Initial program 1.8%

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

      1. add-sqr-sqrt 1.8%

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

      2. pow2 1.8%

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

      3. log-pow 1.8%

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

      4. +-commutative 1.8%

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

      5. hypot-1-def 3.1%

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

   4. Applied egg-rr 3.1%

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

   5. Taylor expanded in x around 0 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 1.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 1.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 17.2%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -44 < x < 43

   1. Initial program 12.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 97.9%

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

   4. Taylor expanded in x around 0 96.1%

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

   if 43 < x

   1. Initial program 56.7%

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

      1. add-sqr-sqrt 56.7%

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

      2. pow2 56.7%

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

      3. log-pow 56.7%

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

      4. +-commutative 56.7%

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

      5. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 11.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 11.1%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 11.1%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around inf 17.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{1}{x}\right)\right) + \left(8 \cdot \frac{1}{x} + 64 \cdot \frac{1}{{x}^{4}}\right)\right) - \frac{21.333333333333332}{{x}^{3}}} \]

   10. Simplified 16.9%

       \[\leadsto \color{blue}{43.666666666666664} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 11.1% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-311}:\\ \;\;\;\;-42.666666666666664\\ \mathbf{else}:\\ \;\;\;\;43.666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-311) -42.666666666666664 43.666666666666664))

C

double code(double x) {
	double tmp;
	if (x <= 5e-311) {
		tmp = -42.666666666666664;
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d-311) then
        tmp = -42.666666666666664d0
    else
        tmp = 43.666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-311) {
		tmp = -42.666666666666664;
	} else {
		tmp = 43.666666666666664;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-311:
		tmp = -42.666666666666664
	else:
		tmp = 43.666666666666664
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-311)
		tmp = -42.666666666666664;
	else
		tmp = 43.666666666666664;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-311)
		tmp = -42.666666666666664;
	else
		tmp = 43.666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-311], -42.666666666666664, 43.666666666666664]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000023e-311

   1. Initial program 7.3%

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

      1. add-sqr-sqrt 7.3%

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

      2. pow2 7.3%

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

      3. log-pow 7.3%

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

      4. +-commutative 7.3%

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

      5. hypot-1-def 7.8%

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

   4. Applied egg-rr 7.8%

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

   5. Taylor expanded in x around 0 5.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.5%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 5.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 10.7%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if 5.00000000000023e-311 < x

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

      3. log-pow 38.6%

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

      4. +-commutative 38.6%

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

      5. hypot-1-def 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in x around 0 10.9%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 10.9%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 10.9%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around inf 10.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{1}{x}\right)\right) + \left(8 \cdot \frac{1}{x} + 64 \cdot \frac{1}{{x}^{4}}\right)\right) - \frac{21.333333333333332}{{x}^{3}}} \]

   10. Simplified 12.2%

       \[\leadsto \color{blue}{43.666666666666664} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 10.9% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;-42.666666666666664\\ \mathbf{else}:\\ \;\;\;\;21.333333333333332\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1e-308) -42.666666666666664 21.333333333333332))

C

double code(double x) {
	double tmp;
	if (x <= -1.1e-308) {
		tmp = -42.666666666666664;
	} else {
		tmp = 21.333333333333332;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d-308)) then
        tmp = -42.666666666666664d0
    else
        tmp = 21.333333333333332d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1e-308) {
		tmp = -42.666666666666664;
	} else {
		tmp = 21.333333333333332;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1e-308:
		tmp = -42.666666666666664
	else:
		tmp = 21.333333333333332
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1e-308)
		tmp = -42.666666666666664;
	else
		tmp = 21.333333333333332;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1e-308)
		tmp = -42.666666666666664;
	else
		tmp = 21.333333333333332;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1e-308], -42.666666666666664, 21.333333333333332]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e-308

   1. Initial program 7.3%

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

      1. add-sqr-sqrt 7.3%

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

      2. pow2 7.3%

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

      3. log-pow 7.3%

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

      4. +-commutative 7.3%

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

      5. hypot-1-def 7.8%

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

   4. Applied egg-rr 7.8%

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

   5. Taylor expanded in x around 0 5.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.5%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 5.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 10.7%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -1.1000000000000001e-308 < x

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

      3. log-pow 38.6%

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

      4. +-commutative 38.6%

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

      5. hypot-1-def 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in x around 0 10.9%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 10.9%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 10.9%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - 8}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 11.7%

       \[\leadsto \color{blue}{21.333333333333332} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 10.4% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-308}:\\ \;\;\;\;-42.666666666666664\\ \mathbf{else}:\\ \;\;\;\;1.2083333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -7e-308) -42.666666666666664 1.2083333333333333))

C

double code(double x) {
	double tmp;
	if (x <= -7e-308) {
		tmp = -42.666666666666664;
	} else {
		tmp = 1.2083333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7d-308)) then
        tmp = -42.666666666666664d0
    else
        tmp = 1.2083333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -7e-308) {
		tmp = -42.666666666666664;
	} else {
		tmp = 1.2083333333333333;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -7e-308:
		tmp = -42.666666666666664
	else:
		tmp = 1.2083333333333333
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -7e-308)
		tmp = -42.666666666666664;
	else
		tmp = 1.2083333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -7e-308)
		tmp = -42.666666666666664;
	else
		tmp = 1.2083333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -7e-308], -42.666666666666664, 1.2083333333333333]

Derivation

1. Split input into 2 regimes

2. if x < -7e-308

   1. Initial program 7.3%

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

      1. add-sqr-sqrt 7.3%

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

      2. pow2 7.3%

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

      3. log-pow 7.3%

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

      4. +-commutative 7.3%

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

      5. hypot-1-def 7.8%

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

   4. Applied egg-rr 7.8%

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

   5. Taylor expanded in x around 0 5.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.5%

         \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

   7. Simplified 5.5%

      \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

   8. Taylor expanded in x around -inf 1.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

   10. Simplified 10.7%

       \[\leadsto \color{blue}{-42.666666666666664} \]

   if -7e-308 < x

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

      3. log-pow 38.6%

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

      4. +-commutative 38.6%

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

      5. hypot-1-def 63.5%

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

   4. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 63.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      2. log1p-undefine 63.5%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\right)\right)} \]

      3. expm1-undefine 63.5%

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

      4. *-commutative 63.5%

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

      5. exp-to-pow 63.5%

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

      6. pow2 63.5%

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

      7. add-sqr-sqrt 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. log1p-define 63.5%

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

      2. sub-neg 63.5%

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

      3. +-commutative 63.5%

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

      4. metadata-eval 63.5%

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

      5. associate-+l+ 63.4%

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

   8. Simplified 63.4%

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

   9. Taylor expanded in x around inf 58.5%

      \[\leadsto \color{blue}{\left(\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0.25 \cdot \frac{1}{{x}^{2}} + 0.052083333333333336 \cdot \frac{1}{{x}^{6}}\right)\right)\right) - \frac{0.09375}{{x}^{4}}} \]

   11. Simplified 10.7%

       \[\leadsto \color{blue}{1.2083333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 6.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -42.666666666666664 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -42.666666666666664)

C

double code(double x) {
	return -42.666666666666664;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -42.666666666666664d0
end function

Java

public static double code(double x) {
	return -42.666666666666664;
}

Python

def code(x):
	return -42.666666666666664

Julia

function code(x)
	return -42.666666666666664
end

MATLAB

function tmp = code(x)
	tmp = -42.666666666666664;
end

Wolfram

code[x_] := -42.666666666666664

Derivation

1. Initial program 23.4%

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

   1. add-sqr-sqrt 23.4%

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

   2. pow2 23.4%

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

   3. log-pow 23.4%

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

   4. +-commutative 23.4%

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

   5. hypot-1-def 36.5%

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

4. Applied egg-rr 36.5%

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

5. Taylor expanded in x around 0 8.3%

   \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 8.3%

      \[\leadsto 2 \cdot \log \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) \]

7. Simplified 8.3%

   \[\leadsto 2 \cdot \log \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot 0.125\right)\right)} \]

8. Taylor expanded in x around -inf 0.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{21.333333333333332 \cdot \frac{1}{{x}^{2}} - \left(8 + \frac{64}{{x}^{3}}\right)}{x} + 2 \cdot \left(\log 0.125 + -2 \cdot \log \left(\frac{-1}{x}\right)\right)} \]

10. Simplified 6.1%

    \[\leadsto \color{blue}{-42.666666666666664} \]
11. Add Preprocessing

Developer Target 1: 30.3% 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 2024131 
(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)))))