Hyperbolic arcsine

Percentage Accurate: 17.3% → 99.7%

Time: 10.9s

Alternatives: 11

Speedup: 207.0×

• could not determine a ground truth

  1. x = -2.4963526948420285e+174

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: 17.3% 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0142:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0142)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 1.26)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
         0.16666666666666666))))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0142) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 1.26) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0142) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 1.26) {
		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.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0142:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 1.26:
		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.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0142)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 1.26)
		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 = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0142)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 1.26)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + ((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0142], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.26], 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[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.014200000000000001

   1. Initial program 5.6%

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

      1. sqr-neg 5.6%

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

      2. +-commutative 5.6%

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

      3. sqr-neg 5.6%

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

      4. hypot-1-def 6.8%

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

   3. Simplified 6.8%

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

      1. flip-+ 7.2%

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

      2. clear-num 7.2%

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

      3. log-div 5.7%

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

      4. metadata-eval 5.7%

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

      5. pow2 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      6. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      7. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right) \]

      8. add-sqr-sqrt 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right) \]

      9. +-commutative 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right) \]

      10. fma-define 6.1%

          \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}}\right) \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 6.1%

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

      2. div-sub 6.1%

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

      3. fma-undefine 6.1%

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

      4. unpow2 6.1%

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

      5. associate--r+ 6.1%

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

      6. +-inverses 6.1%

         \[\leadsto -\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right) \]

      7. metadata-eval 6.1%

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

      8. *-rgt-identity 6.1%

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

      9. associate-/l* 6.1%

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

      10. metadata-eval 6.1%

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

      11. fma-undefine 6.1%

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

      12. unpow2 6.1%

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

      13. associate--r+ 53.8%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. associate-/l* 99.9%

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

      18. metadata-eval 99.9%

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

      19. *-commutative 99.9%

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

      20. neg-mul-1 99.9%

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

   8. Simplified 99.9%

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

   if -0.014200000000000001 < x < 1.26000000000000001

   1. Initial program 8.9%

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

      1. sqr-neg 8.9%

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

      2. +-commutative 8.9%

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

      3. sqr-neg 8.9%

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

      4. hypot-1-def 9.0%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.26000000000000001 < x

   1. Initial program 48.3%

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

      1. sqr-neg 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0142:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0009)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0012)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (* 2.0 (log (sqrt (+ x (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0009) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0012) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = 2.0 * log(sqrt((x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0009) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0012) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = 2.0 * Math.log(Math.sqrt((x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0009:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0012:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = 2.0 * math.log(math.sqrt((x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0009)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0012)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(2.0 * log(sqrt(Float64(x + hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0009)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0012)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = 2.0 * log(sqrt((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0009], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0012], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999998e-4

   1. Initial program 5.6%

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

      1. sqr-neg 5.6%

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

      2. +-commutative 5.6%

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

      3. sqr-neg 5.6%

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

      4. hypot-1-def 6.8%

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

   3. Simplified 6.8%

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

      1. flip-+ 7.2%

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

      2. clear-num 7.2%

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

      3. log-div 5.7%

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

      4. metadata-eval 5.7%

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

      5. pow2 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      6. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      7. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right) \]

      8. add-sqr-sqrt 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right) \]

      9. +-commutative 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right) \]

      10. fma-define 6.1%

          \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}}\right) \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 6.1%

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

      2. div-sub 6.1%

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

      3. fma-undefine 6.1%

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

      4. unpow2 6.1%

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

      5. associate--r+ 6.1%

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

      6. +-inverses 6.1%

         \[\leadsto -\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right) \]

      7. metadata-eval 6.1%

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

      8. *-rgt-identity 6.1%

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

      9. associate-/l* 6.1%

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

      10. metadata-eval 6.1%

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

      11. fma-undefine 6.1%

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

      12. unpow2 6.1%

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

      13. associate--r+ 53.8%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. associate-/l* 99.9%

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

      18. metadata-eval 99.9%

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

      19. *-commutative 99.9%

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

      20. neg-mul-1 99.9%

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

   8. Simplified 99.9%

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

   if -8.9999999999999998e-4 < x < 0.00119999999999999989

   1. Initial program 8.2%

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

      1. sqr-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. sqr-neg 8.2%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 0.00119999999999999989 < x

   1. Initial program 48.9%

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

      1. sqr-neg 48.9%

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

      2. +-commutative 48.9%

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

      3. sqr-neg 48.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. log-pow 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0009)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.00115)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (+ 1.0 (+ (log (+ x (hypot 1.0 x))) -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0009) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = 1.0 + (log((x + hypot(1.0, x))) + -1.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0009) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = 1.0 + (Math.log((x + Math.hypot(1.0, x))) + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0009:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.00115:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = 1.0 + (math.log((x + math.hypot(1.0, x))) + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0009)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.00115)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(1.0 + Float64(log(Float64(x + hypot(1.0, x))) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0009)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.00115)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = 1.0 + (log((x + hypot(1.0, x))) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0009], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.00115], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999998e-4

   1. Initial program 5.6%

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

      1. sqr-neg 5.6%

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

      2. +-commutative 5.6%

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

      3. sqr-neg 5.6%

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

      4. hypot-1-def 6.8%

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

   3. Simplified 6.8%

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

      1. flip-+ 7.2%

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

      2. clear-num 7.2%

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

      3. log-div 5.7%

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

      4. metadata-eval 5.7%

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

      5. pow2 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      6. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      7. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right) \]

      8. add-sqr-sqrt 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right) \]

      9. +-commutative 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right) \]

      10. fma-define 6.1%

          \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}}\right) \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 6.1%

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

      2. div-sub 6.1%

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

      3. fma-undefine 6.1%

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

      4. unpow2 6.1%

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

      5. associate--r+ 6.1%

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

      6. +-inverses 6.1%

         \[\leadsto -\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right) \]

      7. metadata-eval 6.1%

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

      8. *-rgt-identity 6.1%

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

      9. associate-/l* 6.1%

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

      10. metadata-eval 6.1%

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

      11. fma-undefine 6.1%

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

      12. unpow2 6.1%

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

      13. associate--r+ 53.8%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. associate-/l* 99.9%

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

      18. metadata-eval 99.9%

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

      19. *-commutative 99.9%

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

      20. neg-mul-1 99.9%

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

   8. Simplified 99.9%

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

   if -8.9999999999999998e-4 < x < 0.00115

   1. Initial program 8.2%

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

      1. sqr-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. sqr-neg 8.2%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 0.00115 < x

   1. Initial program 48.9%

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

      1. sqr-neg 48.9%

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

      2. +-commutative 48.9%

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

      3. sqr-neg 48.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-undefine 98.1%

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

      3. log1p-undefine 98.1%

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

      4. rem-exp-log 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.00115)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00115:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00115)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00115)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00115], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $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 < -1.25

   1. Initial program 4.2%

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

      1. sqr-neg 4.2%

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

      2. +-commutative 4.2%

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

      3. sqr-neg 4.2%

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

      4. hypot-1-def 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.25 < x < 0.00115

   1. Initial program 9.0%

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

      1. sqr-neg 9.0%

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

      2. +-commutative 9.0%

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

      3. sqr-neg 9.0%

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

      4. hypot-1-def 9.0%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. *-lft-identity 99.5%

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

      3. associate-*l* 99.5%

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

      4. unpow2 99.5%

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

      5. unpow3 99.5%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 0.00115 < x

   1. Initial program 48.9%

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

      1. sqr-neg 48.9%

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

      2. +-commutative 48.9%

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

      3. sqr-neg 48.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0009)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.00115)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0009) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0009) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0009:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.00115:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0009)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.00115)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0009)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.00115)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0009], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.00115], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $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 < -8.9999999999999998e-4

   1. Initial program 5.6%

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

      1. sqr-neg 5.6%

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

      2. +-commutative 5.6%

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

      3. sqr-neg 5.6%

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

      4. hypot-1-def 6.8%

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

   3. Simplified 6.8%

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

      1. flip-+ 7.2%

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

      2. clear-num 7.2%

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

      3. log-div 5.7%

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

      4. metadata-eval 5.7%

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

      5. pow2 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      6. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      7. hypot-1-def 5.7%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right) \]

      8. add-sqr-sqrt 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right) \]

      9. +-commutative 6.1%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right) \]

      10. fma-define 6.1%

          \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}}\right) \]

   6. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 6.1%

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

      2. div-sub 6.1%

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

      3. fma-undefine 6.1%

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

      4. unpow2 6.1%

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

      5. associate--r+ 6.1%

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

      6. +-inverses 6.1%

         \[\leadsto -\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right) \]

      7. metadata-eval 6.1%

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

      8. *-rgt-identity 6.1%

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

      9. associate-/l* 6.1%

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

      10. metadata-eval 6.1%

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

      11. fma-undefine 6.1%

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

      12. unpow2 6.1%

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

      13. associate--r+ 53.8%

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

      14. +-inverses 99.9%

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

      15. metadata-eval 99.9%

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

      16. *-rgt-identity 99.9%

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

      17. associate-/l* 99.9%

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

      18. metadata-eval 99.9%

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

      19. *-commutative 99.9%

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

      20. neg-mul-1 99.9%

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

   8. Simplified 99.9%

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

   if -8.9999999999999998e-4 < x < 0.00115

   1. Initial program 8.2%

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

      1. sqr-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. sqr-neg 8.2%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 0.00115 < x

   1. Initial program 48.9%

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

      1. sqr-neg 48.9%

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

      2. +-commutative 48.9%

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

      3. sqr-neg 48.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.26)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.26d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.26:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.26)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.26)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.26], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 4.2%

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

      1. sqr-neg 4.2%

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

      2. +-commutative 4.2%

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

      3. sqr-neg 4.2%

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

      4. hypot-1-def 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.25 < x < 1.26000000000000001

   1. Initial program 9.7%

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

      1. sqr-neg 9.7%

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

      2. +-commutative 9.7%

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

      3. sqr-neg 9.7%

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

      4. hypot-1-def 9.7%

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

   3. Simplified 9.7%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. distribute-rgt-in 99.2%

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

      2. *-lft-identity 99.2%

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

      3. associate-*l* 99.2%

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

      4. unpow2 99.2%

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

      5. unpow3 99.2%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 99.2%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 1.26000000000000001 < x

   1. Initial program 48.3%

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

      1. sqr-neg 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{1 + \left(-1 + x \cdot 2\right)}{2 - x}\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (/ (+ 1.0 (+ -1.0 (* x 2.0))) (- 2.0 x))
   (if (<= x 1.05) (* (* x (+ x 2.0)) (+ 0.5 (* x -0.25))) (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x);
	} else if (x <= 1.05) {
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = (1.0d0 + ((-1.0d0) + (x * 2.0d0))) / (2.0d0 - x)
    else if (x <= 1.05d0) then
        tmp = (x * (x + 2.0d0)) * (0.5d0 + (x * (-0.25d0)))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x);
	} else if (x <= 1.05) {
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x)
	elif x <= 1.05:
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = Float64(Float64(1.0 + Float64(-1.0 + Float64(x * 2.0))) / Float64(2.0 - x));
	elseif (x <= 1.05)
		tmp = Float64(Float64(x * Float64(x + 2.0)) * Float64(0.5 + Float64(x * -0.25)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x);
	elseif (x <= 1.05)
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[(N[(1.0 + N[(-1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 4.2%

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

      1. sqr-neg 4.2%

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

      2. +-commutative 4.2%

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

      3. sqr-neg 4.2%

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

      4. hypot-1-def 5.4%

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

   3. Simplified 5.4%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

      3. log1p-undefine 0.0%

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

      4. rem-exp-log 5.4%

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

   6. Applied egg-rr 5.4%

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

   7. Taylor expanded in x around 0 5.6%

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

      1. associate--l+ 5.6%

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

      2. flip-+ 5.3%

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

      3. metadata-eval 5.3%

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

      4. sub-neg 5.3%

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

      5. metadata-eval 5.3%

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

      6. sub-neg 5.3%

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

      7. metadata-eval 5.3%

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

      8. sub-neg 5.3%

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

      9. metadata-eval 5.3%

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

   9. Applied egg-rr 5.3%

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

       1. sub-neg 5.3%

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

       2. distribute-rgt-neg-in 5.3%

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

       3. distribute-neg-in 5.3%

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

       4. metadata-eval 5.3%

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

       5. +-commutative 5.3%

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

       6. sub-neg 5.3%

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

       7. +-commutative 5.3%

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

       8. associate--r+ 5.3%

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

       9. metadata-eval 5.3%

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

   11. Simplified 5.3%

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

   12. Taylor expanded in x around 0 14.5%

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

   if -1.25 < x < 1.05000000000000004

   1. Initial program 9.7%

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

      1. sqr-neg 9.7%

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

      2. +-commutative 9.7%

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

      3. sqr-neg 9.7%

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

      4. hypot-1-def 9.7%

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

   3. Simplified 9.7%

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

      1. expm1-log1p-u 9.7%

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

      2. expm1-undefine 9.7%

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

      3. log1p-undefine 9.7%

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

      4. rem-exp-log 9.7%

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

   6. Applied egg-rr 9.7%

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

   7. Taylor expanded in x around 0 8.6%

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

      1. flip-- 8.6%

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

      2. div-inv 8.6%

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

      3. metadata-eval 8.6%

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

      4. difference-of-sqr-1 8.6%

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

      5. +-commutative 8.6%

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

      6. associate-+l+ 8.6%

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

      7. metadata-eval 8.6%

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

      8. add-exp-log 8.6%

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

      9. expm1-define 8.6%

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

      10. log1p-define 98.4%

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

      11. expm1-log1p-u 98.4%

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

      12. +-commutative 98.4%

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

      13. associate-+l+ 98.4%

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

      14. metadata-eval 98.4%

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

   9. Applied egg-rr 98.4%

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

   10. Taylor expanded in x around 0 98.4%

       \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)} \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right) \]

   12. Simplified 98.4%

       \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 48.3%

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

      1. sqr-neg 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{1 + \left(-1 + x \cdot 2\right)}{2 - x}\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (/ -0.5 x))
   (if (<= x 1.05) (* (* x (+ x 2.0)) (+ 0.5 (* x -0.25))) (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.05d0) then
        tmp = (x * (x + 2.0d0)) * (0.5d0 + (x * (-0.25d0)))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log((-0.5 / x))
	elif x <= 1.05:
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.05)
		tmp = Float64(Float64(x * Float64(x + 2.0)) * Float64(0.5 + Float64(x * -0.25)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log((-0.5 / x));
	elseif (x <= 1.05)
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 4.2%

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

      1. sqr-neg 4.2%

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

      2. +-commutative 4.2%

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

      3. sqr-neg 4.2%

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

      4. hypot-1-def 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.1000000000000001 < x < 1.05000000000000004

   1. Initial program 9.7%

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

      1. sqr-neg 9.7%

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

      2. +-commutative 9.7%

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

      3. sqr-neg 9.7%

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

      4. hypot-1-def 9.7%

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

   3. Simplified 9.7%

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

      1. expm1-log1p-u 9.7%

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

      2. expm1-undefine 9.7%

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

      3. log1p-undefine 9.7%

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

      4. rem-exp-log 9.7%

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

   6. Applied egg-rr 9.7%

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

   7. Taylor expanded in x around 0 8.6%

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

      1. flip-- 8.6%

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

      2. div-inv 8.6%

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

      3. metadata-eval 8.6%

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

      4. difference-of-sqr-1 8.6%

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

      5. +-commutative 8.6%

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

      6. associate-+l+ 8.6%

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

      7. metadata-eval 8.6%

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

      8. add-exp-log 8.6%

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

      9. expm1-define 8.6%

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

      10. log1p-define 98.4%

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

      11. expm1-log1p-u 98.4%

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

      12. +-commutative 98.4%

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

      13. associate-+l+ 98.4%

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

      14. metadata-eval 98.4%

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

   9. Applied egg-rr 98.4%

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

   10. Taylor expanded in x around 0 98.4%

       \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)} \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right) \]

   12. Simplified 98.4%

       \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 48.3%

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

      1. sqr-neg 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 + \left(-1 + x \cdot 2\right)}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -9.5e-7)
   (/ (+ 1.0 (+ -1.0 (* x 2.0))) (- 2.0 x))
   (/ (* x 2.0) (+ x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -9.5e-7) {
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x);
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-9.5d-7)) then
        tmp = (1.0d0 + ((-1.0d0) + (x * 2.0d0))) / (2.0d0 - x)
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -9.5e-7) {
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x);
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -9.5e-7:
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x)
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -9.5e-7)
		tmp = Float64(Float64(1.0 + Float64(-1.0 + Float64(x * 2.0))) / Float64(2.0 - x));
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -9.5e-7)
		tmp = (1.0 + (-1.0 + (x * 2.0))) / (2.0 - x);
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -9.5e-7], N[(N[(1.0 + N[(-1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000001e-7

   1. Initial program 6.7%

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

      1. sqr-neg 6.7%

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

      2. +-commutative 6.7%

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

      3. sqr-neg 6.7%

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

      4. hypot-1-def 7.8%

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

   3. Simplified 7.8%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 2.6%

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

      3. log1p-undefine 2.7%

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

      4. rem-exp-log 7.9%

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

   6. Applied egg-rr 7.9%

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

   7. Taylor expanded in x around 0 7.0%

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

      1. associate--l+ 7.0%

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

      2. flip-+ 6.7%

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

      3. metadata-eval 6.7%

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

      4. sub-neg 6.7%

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

      5. metadata-eval 6.7%

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

      6. sub-neg 6.7%

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

      7. metadata-eval 6.7%

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

      8. sub-neg 6.7%

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

      9. metadata-eval 6.7%

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

   9. Applied egg-rr 6.7%

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

       1. sub-neg 6.7%

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

       2. distribute-rgt-neg-in 6.7%

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

       3. distribute-neg-in 6.7%

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

       4. metadata-eval 6.7%

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

       5. +-commutative 6.7%

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

       6. sub-neg 6.7%

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

       7. +-commutative 6.7%

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

       8. associate--r+ 6.7%

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

       9. metadata-eval 6.7%

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

   11. Simplified 6.7%

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

   12. Taylor expanded in x around 0 15.1%

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

   if -9.5000000000000001e-7 < x

   1. Initial program 24.2%

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

      1. sqr-neg 24.2%

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

      2. +-commutative 24.2%

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

      3. sqr-neg 24.2%

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

      4. hypot-1-def 44.5%

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

   3. Simplified 44.5%

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

      1. expm1-log1p-u 43.8%

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

      2. expm1-undefine 43.8%

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

      3. log1p-undefine 43.8%

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

      4. rem-exp-log 44.5%

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

   6. Applied egg-rr 44.5%

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

   7. Taylor expanded in x around 0 6.8%

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

      1. flip-- 6.7%

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

      2. metadata-eval 6.7%

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

      3. difference-of-sqr-1 6.7%

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

      4. +-commutative 6.7%

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

      5. associate-+l+ 6.7%

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

      6. metadata-eval 6.7%

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

      7. add-exp-log 6.7%

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

      8. expm1-define 6.7%

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

      9. log1p-define 62.0%

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

      10. expm1-log1p-u 62.0%

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

      11. +-commutative 62.0%

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

      12. associate-+l+ 62.0%

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

      13. metadata-eval 62.0%

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

   9. Applied egg-rr 62.0%

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

   10. Taylor expanded in x around 0 65.1%

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

       1. *-commutative 65.1%

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

   12. Simplified 65.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 + \left(-1 + x \cdot 2\right)}{2 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.82:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.82) x (/ (* x 2.0) (+ x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.82) {
		tmp = x;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.82d0) then
        tmp = x
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.82) {
		tmp = x;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.82:
		tmp = x
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.82)
		tmp = x;
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.82)
		tmp = x;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.82], x, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.82000000000000006

   1. Initial program 7.8%

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

      1. sqr-neg 7.8%

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

      2. +-commutative 7.8%

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

      3. sqr-neg 7.8%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 65.6%

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

   if 1.82000000000000006 < x

   1. Initial program 48.3%

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

      1. sqr-neg 48.3%

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

      2. +-commutative 48.3%

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

      3. sqr-neg 48.3%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 98.2%

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

      2. expm1-undefine 98.1%

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

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 5.3%

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

      1. flip-- 4.9%

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

      2. metadata-eval 4.9%

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

      3. difference-of-sqr-1 4.9%

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

      4. +-commutative 4.9%

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

      5. associate-+l+ 4.9%

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

      6. metadata-eval 4.9%

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

      7. add-exp-log 4.9%

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

      8. expm1-define 4.9%

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

      9. log1p-define 4.9%

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

      10. expm1-log1p-u 4.9%

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

      11. +-commutative 4.9%

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

      12. associate-+l+ 4.9%

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

      13. metadata-eval 4.9%

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

   9. Applied egg-rr 4.9%

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

   10. Taylor expanded in x around 0 14.4%

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

       1. *-commutative 14.4%

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

   12. Simplified 14.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.82:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 19.6%

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

   1. sqr-neg 19.6%

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

   2. +-commutative 19.6%

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

   3. sqr-neg 19.6%

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

   4. hypot-1-def 35.1%

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

3. Simplified 35.1%

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

5. Taylor expanded in x around 0 47.9%

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

6. Final simplification 47.9%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 28.9% 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 2024073 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

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