Hyperbolic arcsine

Percentage Accurate: 17.6% → 99.9%

Time: 8.1s

Alternatives: 9

Speedup: 207.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.021)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0225)
     (+
      x
      (*
       (fma
        (pow x 2.0)
        (fma (pow x 2.0) -0.044642857142857144 0.075)
        -0.16666666666666666)
       (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.021) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0225) {
		tmp = x + (fma(pow(x, 2.0), fma(pow(x, 2.0), -0.044642857142857144, 0.075), -0.16666666666666666) * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.021)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0225)
		tmp = Float64(x + Float64(fma((x ^ 2.0), fma((x ^ 2.0), -0.044642857142857144, 0.075), -0.16666666666666666) * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.021], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0225], N[(x + N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * -0.044642857142857144 + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * 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 < -0.0210000000000000013

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. sqr-neg 3.3%

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

      4. hypot-1-def 4.4%

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

   3. Simplified 4.4%

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

      1. flip-+ 3.0%

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

      2. frac-2neg 3.0%

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

      3. log-div 3.0%

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

      4. pow2 3.0%

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

      5. hypot-1-def 3.0%

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

      6. hypot-1-def 3.0%

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

      7. add-sqr-sqrt 3.0%

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

      8. +-commutative 3.0%

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

      9. fma-define 3.0%

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

   6. Applied egg-rr 3.0%

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

      1. fma-undefine 3.0%

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

      2. unpow2 3.0%

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

      3. associate--r+ 54.1%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.0210000000000000013 < x < 0.022499999999999999

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. hypot-1-def 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. fmm-def 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-define 100.0%

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

      9. metadata-eval 100.0%

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

      10. unpow2 100.0%

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

      11. unpow3 100.0%

          \[\leadsto x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot {x}^{3}} \]

   if 0.022499999999999999 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.021)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0225)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
         0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.021) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0225) {
		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 + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.021) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0225) {
		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 + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.021:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0225:
		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 + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.021)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0225)
		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 + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.021)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0225)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + ((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.021], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0225], 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 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0210000000000000013

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. sqr-neg 3.3%

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

      4. hypot-1-def 4.4%

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

   3. Simplified 4.4%

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

      1. flip-+ 3.0%

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

      2. frac-2neg 3.0%

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

      3. log-div 3.0%

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

      4. pow2 3.0%

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

      5. hypot-1-def 3.0%

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

      6. hypot-1-def 3.0%

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

      7. add-sqr-sqrt 3.0%

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

      8. +-commutative 3.0%

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

      9. fma-define 3.0%

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

   6. Applied egg-rr 3.0%

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

      1. fma-undefine 3.0%

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

      2. unpow2 3.0%

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

      3. associate--r+ 54.1%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate--r- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.0210000000000000013 < x < 0.022499999999999999

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. hypot-1-def 7.9%

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

   3. Simplified 7.9%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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 0.022499999999999999 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00088)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.00105)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00088) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.00105) {
		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 <= -0.00088) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.00105) {
		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 <= -0.00088:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.00105:
		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 <= -0.00088)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.00105)
		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 <= -0.00088)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.00105)
		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, -0.00088], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.00105], 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 < -8.80000000000000031e-4

   1. Initial program 4.5%

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

      1. sqr-neg 4.5%

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

      2. +-commutative 4.5%

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

      3. sqr-neg 4.5%

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

      4. hypot-1-def 5.6%

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

   3. Simplified 5.6%

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

      1. flip-+ 4.2%

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

      2. frac-2neg 4.2%

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

      3. log-div 4.2%

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

      4. pow2 4.2%

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

      5. hypot-1-def 4.2%

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

      6. hypot-1-def 4.2%

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

      7. add-sqr-sqrt 4.2%

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

      8. +-commutative 4.2%

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

      9. fma-define 4.2%

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

   6. Applied egg-rr 4.2%

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

      1. fma-undefine 4.2%

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

      2. unpow2 4.2%

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

      3. associate--r+ 54.6%

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

      4. +-inverses 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-sub0 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate--r- 99.9%

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

      11. neg-sub0 99.9%

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

      12. +-commutative 99.9%

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

      13. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   if -8.80000000000000031e-4 < x < 0.00104999999999999994

   1. Initial program 7.3%

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

      1. sqr-neg 7.3%

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

      2. +-commutative 7.3%

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

      3. sqr-neg 7.3%

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

      4. hypot-1-def 7.3%

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

   3. Simplified 7.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
   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 0.00104999999999999994 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 0.00105)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00105) {
		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 <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00105) {
		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 <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00105:
		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 <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00105)
		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 <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00105)
		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, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00105], 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 < -1.26000000000000001

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. sqr-neg 3.3%

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

      4. hypot-1-def 4.4%

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

   3. Simplified 4.4%

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

   5. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 0.00104999999999999994

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. hypot-1-def 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. *-lft-identity 99.7%

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

      3. associate-*l* 99.7%

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

      4. unpow2 99.7%

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

      5. unpow3 99.7%

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

   7. Simplified 99.7%

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

   if 0.00104999999999999994 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.26], 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[(0.5 / x), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. sqr-neg 3.3%

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

      4. hypot-1-def 4.4%

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

   3. Simplified 4.4%

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

   5. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 1.26000000000000001

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. hypot-1-def 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. *-lft-identity 99.7%

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

      3. associate-*l* 99.7%

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

      4. unpow2 99.7%

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

      5. unpow3 99.7%

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

   7. Simplified 99.7%

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

   if 1.26000000000000001 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 3.2%

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

      2. frac-2neg 3.2%

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

      3. log-div 3.2%

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

      4. pow2 3.2%

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

      5. hypot-1-def 3.2%

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

      6. hypot-1-def 3.2%

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

      7. add-sqr-sqrt 3.2%

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

      8. +-commutative 3.2%

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

      9. fma-define 3.2%

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

   6. Applied egg-rr 3.2%

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

      1. fma-undefine 3.2%

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

      2. unpow2 3.2%

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

      3. associate--r+ 4.6%

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

      4. +-inverses 6.2%

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

      5. metadata-eval 6.2%

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

      6. metadata-eval 6.2%

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

      7. metadata-eval 6.2%

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

      8. neg-sub0 6.2%

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

      9. neg-sub0 6.2%

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

      10. associate--r- 6.2%

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

      11. neg-sub0 6.2%

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

      12. +-commutative 6.2%

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

      13. sub-neg 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in x around inf 98.1%

      \[\leadsto -\log \color{blue}{\left(\frac{0.5}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26) (log (/ -0.5 x)) (if (<= x 1.26) x (- (log (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = -log((0.5 / x));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = -Math.log((0.5 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 1.26:
		tmp = x
	else:
		tmp = -math.log((0.5 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.26)
		tmp = x;
	else
		tmp = Float64(-log(Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 1.26)
		tmp = x;
	else
		tmp = -log((0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.26], x, (-N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. sqr-neg 3.3%

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

      4. hypot-1-def 4.4%

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

   3. Simplified 4.4%

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

   5. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 1.26000000000000001

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. hypot-1-def 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 99.6%

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

   if 1.26000000000000001 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 3.2%

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

      2. frac-2neg 3.2%

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

      3. log-div 3.2%

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

      4. pow2 3.2%

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

      5. hypot-1-def 3.2%

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

      6. hypot-1-def 3.2%

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

      7. add-sqr-sqrt 3.2%

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

      8. +-commutative 3.2%

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

      9. fma-define 3.2%

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

   6. Applied egg-rr 3.2%

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

      1. fma-undefine 3.2%

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

      2. unpow2 3.2%

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

      3. associate--r+ 4.6%

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

      4. +-inverses 6.2%

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

      5. metadata-eval 6.2%

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

      6. metadata-eval 6.2%

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

      7. metadata-eval 6.2%

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

      8. neg-sub0 6.2%

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

      9. neg-sub0 6.2%

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

      10. associate--r- 6.2%

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

      11. neg-sub0 6.2%

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

      12. +-commutative 6.2%

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

      13. sub-neg 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in x around inf 98.1%

      \[\leadsto -\log \color{blue}{\left(\frac{0.5}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26) (log (/ -0.5 x)) (if (<= x 1.26) x (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 1.26:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. sqr-neg 3.3%

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

      4. hypot-1-def 4.4%

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

   3. Simplified 4.4%

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

   5. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 1.26000000000000001

   1. Initial program 7.9%

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

      1. sqr-neg 7.9%

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

      2. +-commutative 7.9%

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

      3. sqr-neg 7.9%

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

      4. hypot-1-def 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0 99.6%

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

   if 1.26000000000000001 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

Alternative 8: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.26:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.26) x (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.26) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.26d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.26:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.26)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.26)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.26], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.26000000000000001

   1. Initial program 6.2%

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

      1. sqr-neg 6.2%

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

      2. +-commutative 6.2%

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

      3. sqr-neg 6.2%

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

      4. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in x around 0 66.0%

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

   if 1.26000000000000001 < x

   1. Initial program 52.6%

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

      1. sqr-neg 52.6%

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

      2. +-commutative 52.6%

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

      3. sqr-neg 52.6%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

Alternative 9: 52.3% 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 15.1%

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

   1. sqr-neg 15.1%

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

   2. +-commutative 15.1%

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

   3. sqr-neg 15.1%

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

   4. hypot-1-def 24.5%

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

3. Simplified 24.5%

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

5. Taylor expanded in x around 0 54.4%

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

Developer Target 1: 29.4% 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 2024170 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

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

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